What shock do you have ?

Second advice about getting a shock pump and checking your sag. Also what about the rebound settings ? Too little rebound damping may have you boinging a bit ( I think I've got that the right way round)

How is your pedal stroke ?

You do have to spin smoothly on a full-susser else it can feel bouncy on climbs, especially compared to a hardtail. I used to think my Titus (MotoLite) climbed worse than my hardtail as I could feel it "squat" and bounce which bugged me, turns out I wasn't really pedalling smoothly and would try to push a higher gear than I really should, I would get away with it on the hardtail as there is no pedal feedback. If I concentrate on spinning smoothly it climbs like a demon and doesn't squat.