I'm pretty certain that every human being has felt exactly what you are describing at one point or another, albeit some a lot more than others, as feelings of grief, anxiety, and loss effect every person differently than the next. Anyone who says that thinking about the prospect of losing those that they love doesn't effect them emotionally or psychologically is either lying, or they are in denial. Even the coldest or people worry about losing those they love forever.
And while it's completely normal and common for people to have anxiety about possibly losing those they love, the fact that it's impacting you so greatly right now is because you just did lose someone that you loved, a member of your family. It's still fresh, the wound is still raw. The only cure for the grief of losing someone is time, and time doesn't even make that fear and anxiety of it happening again go away, and again, that's normal because you love your family.
When I was 26 all 4 of my grandparents were still living. I remember friends that were the same age as I was telling me how lucky I was that I still had all my grandparents at 26 years old, and I understood why they said this to me, because most of them had lost their grandparents years prior. However, I tried so hard not to think about that situation because I knew what was going to happen and it drove me insane, so much so that I would often have dreams that they had died, then I would wake-up in a sweat and say "Just a dream, they're still alive", and exhale....Then 2 days after Christmas, December 27th, my mom's mother passed-away. I took a few days off from work (which for a car salesperson/manager is impossible, doesn't matter who died or what illness you have, to the point that when I was going through laser treatments and surgeries for cancer I got yelled at on a daily basis and was called "part-time" by the General Sales Manager)...Then on February 1st my mom's father passed away. He was in great physical health for 92 years old (he had severe dementia and only recognized my grandmother and myself at that point, so they actually didn't even tell him that she has passed away, he wouldn't have remembered anyway). We're certain that he passed-away because my grandmother had died, even though he didn't know that she had died just a month prior...So again I took a few days off from work, got hell, and had to deal with the fact that this was happening all at once...Then on March 3rd my dad's mother passed-away, no joke...It was exactly what I had known was going to happen, I lost them all at the same time (though this was a little more brutal than I had ever guessed that it would have been)...they didn't even believe me at work, not my cold-hearted, arrogant, narcissistic boss anyway, I actually had to point out the obituaries in the newspaper, it was disgusting...but what I was feeling was just this feeling of fear, anxiety, loss, grief, and frustration, along with a huge, empty hole in my life all of a sudden. My dad's father lived for another 8 years after that, so I didn't lose all 4 of them at the same time, but losing 3 of them within 3 months was a bit much for me to take. And I remember walking around through life constantly wondering when he was going to die. Same thing happened with the first 2 dogs I ever owned on my own, my first Australian Cattle Dog, Cleo, and my Pug, Jaeger. I got them about 3 years apart, but Cleo was born with a heart murmur that caused no issues until she hit about 10 years old and started retaining fluid. I had her belly drained several times, but then at the age of 12 she started to have seizures whenever she would lay-down for a long period of time and then suddenly stand up, as her blood-pressure would rapidly fall due to the fluid and the weakness of her heart not pumping strongly enough. I had to put her to sleep just before her 13th birthday. Jaeger the pug was about 10 then, and I came home from work one night to find that he had died in his sleep on the couch about a year later, very young for a pug as he had just turned 12. My house was empty, so I did shortly after that go and find my current Blue Heeler, Lola, who is now 4 years old, very shortly after Jaeger died. I only had my breeder Budgies at the time as far as parrots, so it was very hard.
I guess for me, I'd rather feel those emotions of anxiety and fear of losing my family than not, as I said they do and will lessen for you, as they are still very, very fresh and raw for you right now. If we didn't worry about losing our loved ones then they wouldn't be loved ones, or at least not very important loved ones. And even though this current constant anxiety, fear, and grief that you're feeling will definitely lessen with time, it is true that we could all go at any second. We could be killed in a car accident on our way to work, but that doesn't mean we're going to stop leaving the house. I'm an avid mountain-biker, I've been doing it on a regular basis since I was 12 years old, and I bike on some very nasty but fun trails. And I've had a few very bad wrecks, one in-particular where I rode right off the edge and into the side of a cement drainage ditch that was covered over with shrubbery that totally hid it. I had a concussion and laid in that ditch for hours before I was able to crawl out of it and walk my bike up to the rode for help...I could have easily died, but that was about 15 years ago, and I haven't stopped biking, because I love it.
The point is that while it's totally normal to fear losing people and animals that you love, if you let that anxiety completely overwhelm you, then you'll end-up not living anymore. You can do everything you can to protect your birds and keep them safe, feed them a very healthy, varied diet every day, give them lots of exercise, keep them separated from any other animals/pets, and make sure that you take every precaution you can so that they don't fly away outside, such as always checking doors and windows, and always having a harness and leash on them or having them in a carrier when you take them out of the house, etc., but even all of these things that I just typed out seem like a lot to think of every single day, every second of the day. You'll end-up not enjoying your birds if you only feel anxiety about losing them instead of feeling the joy that they bring you when you spend time with them. There's nothing any of us can do to ensure that nothing will ever happen to our pets, we can't control that, but what we can all control is the quality of life that we give them and the amount of love that we give them while they're here and while we're here.
One more thing that I hope you'll keep an open-mind about, as some people won't ever consider this when they are going through something like this, either due to pride or fear, or for whatever reason, but if this anxiety that you're feeling is really consuming your every moment and preventing you from doing things like concentrate at work or even enjoying the time that you spend with your birds, and you don't think that it's getting any better soon, it's important to remember that it's always a very good idea to talk to someone about it. I don't know if you've spoken to any of your family, friends, significant-other, etc. about what you've been feeling, but if not, that's a really good place to start. A lot of the time just saying the words out-loud, just telling someone else what you're feeling and relieving yourself of that burden that you've been carrying around inside you is enough to start the healing process and lessen your anxiety. And if you try that and it doesn't help you, then there is absolutely nothing wrong with seeking out the help of a professional. I know personally that talking to someone who is completely neutral, who doesn't know anyone else that I know, that doesn't really know me personally, at least not at-first, someone who will not be judgmental and who is just there to listen to you and nothing more is the best treatment for emotional pain in the world. Something to think about...
