Purchasing a car while PCSing

Hello!
I'm going to be PCSing soon and will be driving in the states and I'm debating on purchasing a car while driving.
My question is if I do purchase a car in another state, which state would I pay taxes on?

I'm currently in a no sales tax state and moving to a sales tax state and may buy a vehicle in a different state.
I don't plan on switching my residence over to the new state.

And If I do pay taxes on which ever state, am I able to claim that in my tax return since I'm still showing residence with my current no sales tax state?

Thank you!