Is there anyway that I can not worry about drill until I am settled into TN? Will I be punished if I stopped showing for drill?

SPC, the best advice I could give you is to maintain constant communication with you leadership. I would email your FFL, 1SG, and UA. Loop them all in on your situation. You are able to request excused absences from BA but your unit will need a good reason, which you seem to have, and you'll need to be 100% on all your readiness, medical, admin, etc.
In order to stay off of the Non-participants list you have to receive pay every 90 days. That can be for drill, LHI appointments, EBDL, or RSTs. Work with your unit to request those days. You can RST during the work week if your unit can use you or you can request to work at the recruiting station or other military organization. This would require pre-authorization from your 1SG and CDR. You can request a memo to present to your employer and your kids would be at school.
Lastly, find your new unit in TN. Log onto the HRC Soldier portal > tools > vacancy search, and log for your MOS. Contact the unit and link them up with your UA or RPAC you get your TPU transfer request packet started. They can state a specific start date of you know when that will be. Your current unit will be a lot more flexible of your transfer order is pending.

Feel free to reach out of you have specific questions.

Be careful with not showing up especially since you’ve stated you received a bonus for this enlistment because you don’t want to put yourself in a position where they recoup the bonus. In order to keep the bonus you’ve received you must fulfill your contractual obligations.

You need to bring these issues up with your Chain stat. Best case scenario is that they allow you to do RSTs each month and you work from home doing EBDL courses that will get you paid for the training and meet the required amount of hours you'd miss for BTA. Worst case is that you don't show up and you start getting U's and end up being put out for getting too many of them.