(Rekordbox user, but may still apply):

Even if "Beat Sync" is on (or BPM sync, whichever you prefer) and the songs sound slightly off, I bring my tempo slider all the way down to 0 BPM and look at where the "beat lines" are in reference to the actual timing of the track. If the computerized beat lines are off from the track's actual timing, I adjust said beat lines to the track and that usually solves the issue. 

[What I mean by 'beat lines'](https://i.imgur.com/f5O5ZWb.png)

[Off tempo 'beat line' at 0 BPM](https://i.imgur.com/Ku7f6Tt.png)

[Correct tempo 'beat line' at 0 BPM](https://i.imgur.com/zWKq52v.png)