I don't know how experienced you are with the XP-60...if you know it like the back of your hand you probably already know about the things I'll say. If not, then check out the following:
I did a song once using the preset Alternative (somewhere in PR-B). I noticed that sometimes this sound was a little out of tune. I checked the envelopes and settings for the actual patch and found out that it's Random Pitch Depth was above zero (I forget how much). This means the pitch was supposed to vary somewhat while playing--that was part of the design. I wasn't thrilled and my solution was to write it to USER with the new version having no Random Pitch Depth. I used the opportunity to tweak the sound to my satisfaction and problem solved.
The other thing you might consider is Pitch Bend changes. You said that when you had a problem, you would have to start the song over. This may be because if you start a song in the middle, the sequencer starts the playing the microscopes at exactly the point you pressed Play. If you had control changes occur before that point in the song, they will not affect the music.
For example:
At measure three you have a pitch bend change to 600.
You have your synth solo starting at measure 3, beat 2. It continues to measure ten.
You press play at measure four.
Your synth solo will be out of tune because the pitch bend occured before you pressed play. The machine will simply play the solo at Pitch Bend zero because it hasn't been told to bend. This is why you would have to start the song over, to make sure all control changes happen when they're supposed to.
Hope this helps.