I didn't pull the cover off the service but I assume the dryer is 20A...

They are usually 30.

I suppose it's just 3 wire, but maybe I could pulle something that
could handle the 220, 110 & ground.

If you can pull a new wire in why are we dancing around. Just put in a bigger
breaker, pull in some fat 4 wire and put a sub panel to serve the dryer and
your shop.
We beat that subject to death a few days ago but we can hit the high points
again if this is a possibility.