Odd. THere are plenty of people (including me) who have floated barrels without running into this. Theoretically, the only thing "holding" the barrel up is the receiver where the barrel is clamped in. I have even been told that removing the band and doing nothing else will inprove groups since the stock will no longer affect the barrel. It is possible that your barrel to receiver fit is loose and causing the barrel to sag now that the stock is no longer holding it up.
Check the tightness of the two hex screws holding the barrel block to the action. If they are loose, that might casue the problem. If they are tight, the barrel might not be fitting into the receiver tight enough.
You could try getting an after-market target barrel. If you do that, then you might as well install a competition trigger kit and get a new stock. You aren't on a budget are you?
ETA: Go to www.rimfirecentral.com Those guys know as much abotu 10/22's as some people here know about AR's.