I haven't ever flown to an airport where there was not an
FBO with a hangar or preheat to bail me out, but now I need to.

Question: Is it enough to wrap the blanket around the cowl and plug in
the Tanis even though the plane is sitting outside with the cold wind?
How often do airports offer this?

Depends on the airport. Call the airport manager per the AOPA
Directory.

For example, SSQ (Shell Lake WI) has a warm building, an outside
accessible outlet, but bring your own extension cords. We have a
posted list of local volunteers to help too.

SSQ manager