Awww...Tuti is adorable! Congratulations!!
We also purchased our Franklin from Petco when he was just 2 1/2 months old. We've had him 1 1/2 years now. The show he put on to get our attention and sucker us into bringing him home has never been topped yet. We've NEVER regretted our purchase, we love him to pieces!
Like your Tuti, he never bit in the store, he obviously took to us immediately and there was no way we could walk out of that store without him.
Once home things were great, he curled up in a blanket with my spouse and took a nap and we ooe'd and awwwe'd over him...we couldn't have been more blessed parronts.
As he got more comfortable in his surroundings, he also became more bold, his "testing" nibbles became all out land basting bites that hurt! He would get irritated over goodness only knows what sometimes and have to seek revenge by fluffing up his feathers and running at us with his beak open in "attack" mode!
Franklin is an all out character!
For several months it was hit or miss if he bit while out of his cage, and it seemed we ALWAYS got bit when offering our finger to let him out...he'd run over to his opened door, stand there and when we'd offer our hand, he'd reach out like he was just going to "test" us and WHAM! He'd nail us with a nearly skin breaking bite.
I read everything I could get my eyes on and applied what I thought would work in our situation, and I came up with some solutions of my own that worked for us. I'll condense what worked for us here.
First off, we had his cage placed at and just above eye level. If he climbed up to the top of his cage, we couldn't reach him very well and he'd evade us and attempt biting. I decided that he thought he was king stuff being above us, so we moved his cage to a lower stand so he was chin level and below. That made a HUGE difference! HUGE!
He still bit a few times, but we didn't jerk away...this is very important when your bird bites...don't jerk away...your bird essentially "wins" when you do and it becomes a game and it's only fun for your Tuti, NOT for you!
We taught Franklin..."Beee Niiice" and when he didn't bite we'd immediately "reward" him with a "Good Boy!!!" and he really liked that...AND he got to be out with us...if he bit us when offering our hand to get him out, we'd just shut the cage door and try again later. Don't get your emotions in the way...You open the cage door, Tuti comes to the door and appears to want out, you offer your hand, Tuti bites, you can say something like...ouch, or no bite...and then simply...shut the door and walk away. Try again later.
This worked like a charm for us, although one week, Franklin didn't get to come out of his cage for 3 days...FINALLY on the end of the 3rd day...I thought, what the heck, lets try again...and I opened the door, he came to the door, he reached out with his beak, very tentatively...when he didn't bite me, I said...goooood booyyyyy...it's okay, come on....he checked and tested and finally stepped up and THAT was our HUGE break through moment.
We've rarely had an issue after that.
Now please don't think we just ignored him for 3 days...we sat with him, talked to him, gave him treats and tried to get him out numerous times a day. But every time he bit, we'd shut the door and walk off...sometimes I'd try again in 10 minutes, sometimes an hour...just depended on what I had going on at the time.
When we have him out, we always have toys for him to play with, sometimes a toy would seriously "tick him off" and he'd fluff up and run at myself or my spouse and yes! He'd attack! Course at first we thought it was so funny, his antic...and of course US cowering away from this little 60 gram bird! But after a few times we decided this wasn't funny and when he'd need to seek revenge for what ever reason he thought he had...as soon as he fluffed up and started running toward us, another bird, our dog...what ever...I'd toss a cloth, t-shirt, towel, blanket...what ever over him and leave it.
He loves to be under things anyway, so for HIM, this wasn't traumatizing...but it broke his concentration and essentially gave him a "time out" for a few moments and he'd sometimes sit under his "cover" for several seconds, up to a couple minutes in some cases, but when he eventually climbed out, he was back to "Nice Frankie" and he'd go about playing like nothing happened.
One of Franklins favorite things to do with us is to burrow under blankets and make tunnels and play "hide and seek" or "Peek a boo"
Those were the 2 biggest behaviorial issues we had with Franklin during the first 6 months we had him. It's been amost a year now and our relationship is AMAZING! No one has been bit by him and if we see his body language change to "aggression" we can tell him...Frankie...beeee niiiccce...in a somewhat of a warning tone, and he actually really does back off and redirects his attention.
He talks, he says Hello, pretty bird, Frank the Tank, Chicken Little (one of our cockatiels name) Hey baby and once in a while he says him a good boy.
Consistency was our best tool. Reading his body language and not doing anything to instigate or encourage his biting.
Franklin is such a pleasure, but it was a bit "rocky" for the first 6 months or so.
Good Luck with Tuti.
Also, for Franklin...some of his FAVORITE treats are apple slices, grapes and banana chips

He's a really good eater now, and loves just about everything we eat that he can have.
Chocolate and Avacado are absolute NO NO's..so are apple seeds actually most all fruit seeds/pits are no no's. Lots of fresh fruit, steamed and raw veggies, cooked pasta, wild and brown rice, sweet potato, squash, quinoa, and many herbs are safe and healthy for Tuti.
Again, congratulations and good luck!
Toni