What happens to Medicare Supplement coverage when moving to a different state?

I’m turning 65 and will apply for Medicare next month when my 6-month enrollment window opens. I plan to choose a Medigap plan but am unsure how to select a carrier. I’ve reviewed Humble Dollar threads, done web research, and consulted Claude, but I’d value your personal experiences and wisdom.

One wrinkle: Our permanent address is in Texas, but we own homes in two states and may relocate in about five years. Should I choose a carrier highly rated in both states, or just where I am residing currently? I’m concerned that changing my permanent address might require new underwriting and, thus, affect my ability to stay in a Medigap plan.

Is the best way to solve this potential issue to choose a carrier highly rated in both states or just focus on my current state? Why or why not?