I think you're overstating just how "obvious" this adverse selection problem is, because we haven't seen this play out historically. It sounds like a purely theoretical problem.
Canada established its single payer system provincially, one-by-one, and they didn't have this adverse selection problem. Saskatchewan was the first Province to offer single payer in 1947, followed by Alberta in 1951, etc. By 1961, all Provinces had some form of a single payer healthcare system.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3110239/
> Taxpayer-funded medicare in Canada did not appear at a single point in time: it emerged over a quarter century from 1962, when physician services were covered across Saskatchewan, to 1987, when the demise of optional “full billing” in Ontario began.
So at any given point in time between 1962 and 1987, there was (in theory) this adverse selection problem among certain provinces; and yet that was never really a major problem in Canada. Even to this day, the Canada Health Transfer which provides block funding to Provinces by the Federal government for the purpose of funding these single payer systems covers ~22% of the cost (https://www.cmaj.ca/content/192/45/E1408).
Additionally, even in present day, the European Union has open borders and free immigration between member states, and certain member states have far more generous welfare systems than others, and you still don't have this adverse selection problem, at least in a manner that meaningfully renders those welfare systems unsustainable. Even domestically, we didn't really see dramatic net migration from states that refused to take the Medicaid expansion to those that did.
The reality is that people don't choose where to live strictly on the basis of which state provides taxpayer funded healthcare. In fact, those that are most likely to do that (poor people) are likely already covered by Medicaid or heavily subsidized ACA plans (https://www.kff.org/policy-watch/millions-of-uninsured-ameri...).
And even if one were to engage with the purely theoretical problem, there are ways to mitigate the adverse selection, e.g. by requiring a minimum number of years of residency in order to qualify. This is what US States do today to establish residency for the purpose of in-state university tuition. Given the plethora of historical (and current) data points that suggests that this isn't really a problem, States can choose to cross that bridge if/when they get there.