My first customer is in another state and now I'm confused

Oh boy. Sales tax is a rabbit hole. Generally you only collect tax in states where you have nexus (physical presence or economic threshold). For one out-of-state sale, you probably don't. But check Texas rules. Probably safer to not charge tax and note it as buyer's responsibility. Not a lawyer.

You might have to collect sales tax if you exceed a certain amount of sales in Texas. But one sale? No. Also, shipping is easy, just use USPS flat rate or Pirate Ship for discounts. Don't overthink it.

GrumpyMooseHat basically covered it. But I'd add: keep records. Even if you don't collect now, if you make more sales to Texas, you may need to register. Each state has different thresholds. Also, You might want to use a platform like Etsy that handles tax for you, but sounds like you're doing it yourself.

For a $200 item, definitely get insurance. USPS priority mail includes $100, so you'd need to add more. Or use UPS/FedEx. And yes, just tell her tax isn't included and she may owe use tax in Texas. Most people don't care.

Actually, you might want to check if you need a sales tax permit in Texas. Some states require it even for one sale if you're shipping into the state. I'm not sure about Texas specifically but it's worth a quick call to their revenue department.

I've been selling online for years and never bothered with permits for one-off sales. The internet makes people paranoid. Just ship it and move on.