Is it normal to be on TDY for AIT (to reclassify) and to have my UIC is reflecting that I’m permanent party here at my AIT station?

You messed up by not getting your orders prior to going to school. It's not like when you join the Army and you find out where you're going when you're in school. You were on assignment and have been for a while. You are attending school in a TDY Enroute status to your next duty assignment. Even if you're supposed to return to clear, you're still TDY Enroute. This is going to cause all kinds of issues for you when it's time to get your travel voucher paid if you never received your orders or your DD1610.

As for the reason your UIC is changed is that the school "arrived" you the same way they would arrive an IET student. This is a mistake on their part and usually caused by doing what you most likely did and showed up without your 1610.

You'll need to go back to your Career Counselor and tell her this once you return:
She will need to contact the Fort Sill MPD and they need to "unarrive" you in emilpo. Your orders and 1610 cannot be processed while you are arrived at Fort Sill. Then you will need to process a deferment for your follow on assignment while you and your CC untangle this Gordian knot. The deferment can't be processed while you're arrived to Sill. Then you need to process your levy packet and turn in that for your DD 1610 to pay for your TDY. Also you'll need to work on your command sponsorship for your family if you're taking them overseas.

If you don't have family or you're going to Korea unaccompanied that's another level of crap to untangle because you aren't authorized to return to clear and were required to have cleared post before you went to school.

Absolutely DO NOT email your future branch manager because they'll only give you the wrong information that will make things worse. The same for the school house. You could call HRC but only RRB section will have the right answers, which I've given here, and refer you to your Career Counselor. If that Counselor is SFC Mizer, you can tell her that she has my sympathies and she can call me with any questions.

SPC (Join to see) SFC Boyd, a Senior Brigade Career Counselor will be able to help you.

SFC (Join to see)

I've had this happen a few times with COVID. Schools that are normally a PCS are TDY and return and it's messing people up. Also, sometimes school houses are pulling students into a UIC IOT accomplish certian admin functions. So if you left Sill on a 1610, call your CoC and your 1SG can go yell at S1 for you droppong off the AAA-162. If you left Sill on what should have been a PCS, your unit may have to clear you from Sill by proxy.