It's not anything you're doing wrong at all...Most Budgies that are sold as pets are not hand-raised as babies by a breeder, they are raised totally by their parent birds, and pretty much all of the ones sold in pet shops, especially the big pet shops, are not even interacted with by a person at all until they get to the pet shop they are shipped to, and they are at least a few months old...So they are essentially "wild" birds when they are originally purchased. That's why they are acting like wild birds, because they are...
Budgies are parrots, so they have the intelligence of a 2-3 year-old human child, and as such, when they are not hand-raised nor even handled or interacted with by a human when they are young babies and they are essentially "wild" birds that were born/raised in captivity, they are going to retain all of their natural instincts of survival, and people are going to be 'predators" until they are hand-tamed by a person...I've hand-tamed dozens of parent-raised, non-tame Budgies that have come from pet shops like Petco, PetSmart, Petland, etc., and on-average it takes between 6 months to over a year of working with them every single day without fail for about 15-30 minutes a day, to get them to the point where they will willingly step-up for you onto your finger. So it's a marathon taming a Budgie, not a sprint.
Here are the best tips I can give you for working on hand-taming your two Budgies (by the way, do you know their genders? Because if it's a male/female pair, you're going to end-up with eggs):
#1) Make sure that you have their cage(s) located in the "main room" of your home, meaning the room of the house where you and the others who live there spend most of their time when they are home. Usually this is the living room, family room, den, TV room, etc. Their cage needs to be located in the room where people spend their time, because just simply being among and around people constantly will help tremendously to ease their general fear of people. If you keep their cage(s) in some spare bedroom or other back room away from "where the action is" in the house, they are never going to stop fearing people.
#2) Unfortunately you got a second Budgie and put them in the same cage together (I think), so you need to really think about separating them into two different cages. There are a bunch of reasons you should put them into their own cages, but if you want to hand-tame a non-tame Budgie or Budgies, then it's going to almost be impossible to do when they are bonded so closely to each other. When given the choice between bonding with another Budgie or a person, they are always going to choose the other Budgie. And the two you have have been living together for months already with no fighting or aggressiveness (I'm guessing they don't fight, if they do then they must be separated), so that means that they like each other and are probably bonded closely already, and the chances of you hand-taming both of them together like this are slim...If you separate them into two different cages then they both are going to be much more apt to bond with the person trying to hand-tame them. Also, if they are a male and a female, then you must separate them because not only will you be getting fertile eggs laid constantly, but the female can be literally "bred to death"...
#3) STOP REACHING YOUR HANDS INTO THEIR CAGES TO TRY TO GET THEM OUT!!! Parrots/birds are very, very territorial, and their cages are their "Safe Space", and they will not EVER want your hands inside trying to grab them. Even the most hand-tame parrot who is closely bonded to their human owner will still give that owner "warning bites" when they stick their hands inside of their cages, even just to feed them and give them water each day...I have a 3 year-old Quaker Parrot who is the sweetest, gentlest, most loving bird in the world, and who is bonded to me like my own child. She flies to me when called, she allows me to hold her, flip her over and scratch/touch her everywhere, sleeps on my shoulder for hours, snuggles, etc. But whenever I put my hands inside of her cage, to this day after having her since she was 11 weeks old, she still gets very upset with me...So I simply open her cage door and let her come out on her own, and then she's fine...So the only time you should EVER put your hands inside of their cage is to quickly change their food and water, and that's it. Don't attempt to touch them inside of their cages right now, and you shouldn't ever try to grab a bird around their body...
#4) You're going to have to start working with them each on a one-on-one basis, and never try to work with them at the same time, and never work with them while the other is in the same room, nor when their cage is in the same room. If you try to hand-tame/train them when they are in the same room they are not going to be interested at all in you and will only try to get back together. And the same goes for their cages, if you work with them in the same room where their cage is located, they will only focus on getting back to their cage...So you need to choose a "neutral" room where you can take their cage, let one of them out into the room, take their cage back outside of that room, and then shut the door, so you only have one bird with no cages in the "Training-Room", and this will keep each bird focused on only what you are doing...It will also remove any "Safe-Spaces" that they will continually try to get back to instead of paying attention to you. It must be a room with a latching door obviously, and it should be a room with carpeting so that you can work with the bird on the floor (soft landings as well). If carpeting isn't an option that's fine, just make sure that you can take one bird at a time into it for Training/Taming sessions and you can take their cage back out of that room after getting them out of their cage and into the room, then shut the door...The best way to do this is to carry the cage into the room and either remove the bottom of the cage and open the door, then put the upper part of the cage on the floor and try to get them down onto the floor, then remove the top part of the cage also from the room.
***It will also help greatly is you have a "Training-Perch/Stand" for them to perch on when doing training sessions, in-fact it's almost essential and required when first starting to train/tame a Budgie. For them you just want a small, tabletop perch that you can just carry to the room easily and put down on the floor or on a table, and this will eventually be the place that they will each go to as soon as they enter the room. You can usually buy the very small, tabletop perches for around $10-$20 at some pet shops, all bird shops, or better yet you can build one from PVC piping and fittings you buy from Lowes very, very cheaply...I'm not talking about a large on that stands on the floor or anything elaborate at all, I'm just talking about a portable little perch that is meant to sit on a tabletop and is only about 1 foot long or so total. They consist of only a single perch that is about a foot long and small in diameter for a Budgie, and it's connected to a base that is flat on the bottom so it can sit on a tabletop, and the whole thing weighs less than a pound. You don't want anything elaborate or distracting or huge, you just need a portable perch for them to sit on while you're working with them.
#5)****This one some people will disagree with, but as someone who has hand-tamed/trained dozens and dozens of non-tame Budgies I can tell you that it will help you tremendously in hand-taming them both; in addition, since you are totally new to owning parrots/birds and you've never hand-tamed one before, and because you've already had them for 6 months and have had issues with them flying all around the room, I think what I'm going to suggest is not only the best option for you, but almost necessary for you to succeed in taming your Budgies...(The worst thing you can do when trying to hand-tame a bird is to chase them all around the room trying to catch them to get them back inside the cage, and then have to grab them and put them back in, every time you do that you set yourself back to the beginning because you're scaring them to death)...My suggestion is that you take both your Budgies to either an Avian Specialist Vet or a bird shop and have both of their wings clipped. This is going to help you so much with hand-taming them, especially since you've had so much trouble trying to get them back inside of their cages. Like I said, every time you chase them around the room and grab them to put them back inside of their cages, you're setting yourself back to the beginning...