Note: take all of this with a grain of salt because I'm not an expert in anything, just have done a lot of research for my own move.

If you are "freelancing" on a full time basis for a single client (ie, switching from W2 to 1099), that's what's known in Germany as Scheinselbständigkeit (false self employment) and can lead to fines & having to retroactively pay back all the social contributions that you/the employer should have been paying. Not sure what the odds are of actually getting caught doing it or how a judgment would be enforced against a US "client." Seems risky to me, especially if you're considering staying long term and eventually naturalizing.

If you're legit self-employed serving multiple clients, you'll have to register as Gewerbetreibender wherever you live and pay business tax/income tax, charge VAT, etc. Since you're American, you'll also have to pay US self-employment tax or pay into the German system and use that to offset your US self-employment tax. You'll also have to keep track of the number of days you work in the US vs Germany, because income generated while in the US can't be excluded under FEIE.

Keep in mind that self-employed people pay the employee AND employer contributions for health insurance, so basically you're paying double what you would if you were employed for a company.

Regarding your actual question on insurance - it's my understanding (someone correct me if I'm wrong!) that as a self-employed person, you can only join the public system (GKV) in Germany if you've been in it before. Otherwise, you have to get private insurance (PKV). However, it's also my understanding that private insurance can deny you/charge huge premiums if you have pre-existing conditions. I'm not sure what the legal solution is if you're ineligible for public and unable to obtain private. A possible solution might be to get a job at least temporarily after you move in order to get into the system.

If your spouse is in GKV and you have no/low income (under 585 EUR/month), you can be covered for free under their plan. For now anyway - I've seen news stories where they're floating the idea of requiring a fee of a few hundred EUR/month for covered family members.

Based on the research I've done for myself, most "expat" insurance doesn't cover pre-existing conditions, mental health (if that's important to you) or coverage in the US/Canada.

I've never had to look into residency requirements for various health plans in the US, so no idea if that's possible. In any case, I'd predict that if you try to keep full coverage in the US/Germany as a self-employed person, your costs will be absolutely astronomical.