Distributions from a HSA can be for the spouse.  Current insurance has no effect on that.

The taxpayer can continue to contribute to the HSA, as long as the taxpayer qualifies.  The fact that spouse does not qualify to contribute does not affect the ability of the taxpayer to contribute. 

If the High Deductible Health Plan covers the spouse, the taxpayer can still contribute the 'family' amount to the HSA.  The fact that the spouse does not qualify to contribute does not change that.