Bicycle Mechanics - bike handleing poorly

I just rebuilt and old bike the headset seems stiff but the LBS says it's OK. The front wheel seems alittle off center and the bike rides erraticly. Could the forks or the frame be bent? I used this bike on a wind trainer during the winter. My new bike is a Giant TCRC2 but I was hoping to use this bike as a beater for the winter.

A too-tight headset will make a bike handle badly. It should only be tight enough such that you don't feel any rocking of the front fork. Set properly, the handlebars should flop from side to side freely.

The front wheel being off-center could also be an issue. First, make sure it's seated in the drop outs properly. Then check to sure your brakes aren't off-center making your wheel look off-center when it's really not. If both of those things are ok, check your stem alignment (you were just messing with the headset). The only problem I can think of would be the dishing being off on the wheel but that seems unlikely. My money is on the stem.

Your headset might be adjusted a bit too tightly, is it a threadless or threaded headset? Like All bearings, you want it just tight enough to eliminate play without crushing the actual bearings. If your wheel looks off center and the bike is handling poorly you may want to make sure that your dropouts on the fork are properly aligned, your LBS should have a tool to do this. Of course if you crashed the bike any time recently it may just be that you bent the steerer which would explain both issues at the same time.