Whoa! Sounds like the dist. bushings are WAY worn to me. Even if it doesn't correct the bog, the distributor surely needs to be replaced. However, with the points conceivably gapped that wide when running, it might cause such poor coil saturation (lousy ignition) that it is indeed causing the problem you report.
When you say there is play in the distributor shaft, exactly how are you pushing? Are you pushing straight sideways against the rotor, or are you turning the rotor, pushing against the side of the distributor, or what?
I'm no distributor expert, but there is some play in there for the weights to adjust the timing. Did you have the arm that opens the points on the top of the cam when you set the gap? If not, then turning the rotor would probably change the gap.
I had points open against cam. At this point if you would turn rotor points would close. I would have gap set at rest against the high point on shaft set screw is tight. Then if you push on shaft not letting points off high spot on shaft, points would open to about .028 gap.
Do Mikie's test for dwell vairiation first. If the dwell jumps around then the bushing is worn. Don't forget that dwell also affects timing as well as coil saturation. What engine setup are we talking about here?
I replaced with new Prestolite electronic type. Boat did not miss a beat with new one. When I got boat ready for launch my obsolete mercruiser distributor had bad wire and shorted out. My buddy had a Volvo distributor laying around so I put it in since the price was right (free). What I think happened is that it ran ok for two trips but some rust on the shaft must have taken out the bushings. The good thing is the new one uses same cap and button as old one so I can use the backups I had bought.