Sounds like the previous owner might have had a problem applying the fundamentals of shooting, and rather than correcting HIS problem, decided to try to fix it by moving the sights. In fairness, the long/heavy DA pull causes a lot of folks problems when shooting this pistol, so off-centered sights are fairly common on these guns, IMO.
For you, if it's shooting right, and the rear sight is offset to the right side of the slide, then all you have to do is move the sight back to the left and whatever group you have should be better centered.
In my experience, .380 caliber guns are VERY ammo-sensitive, both for accuracy and functioning. This basic design pre-dated the Hydra-shock (and indeed, most hollowpoints) by several decades, so I personally wouldn't depend on it to function well with modern JHPs. The FMJs are another thing altogether; a current-manufactured pistol should run with most ANY round-nose jacketed bullet, IMO, but going back to my personal experience with this model, that wasn't the case with the one I owned, either. It ran fine with some FMJ loads, and not well at all with others, so I'd recommend trying some different ammo before passing final judgement on functioning. Even so, mine shot quite well with one load that it didn't like, functioning-wise, but functioned well with another load that shot groups twice as large as the first load (both shot off a rest). One of the reasons I didn't mind selling it.
One more note on the one I owned; it absolutely HAD to be kept relatively clean and lubed. Once ready to fire, I would get about 3-4 mags out of it reliably, then I'd start getting failure-to-chamber stoppages, which could be fixed by a quick cleaning of the chamber. Once swabbed-out and re-lubed, it would be ready for another 3-4 mags before problems began again. I simply could not get more than 3 mags out of it reliably without cleaning the chamber; it seemed to be related to powder granules building-up and interfering with chambering.
I sold mine after describing the difficulties to another shooter, who decided he was up to the challenge of fixing it. I asked him about it at the next annual gun show, but he quickly changed the subject, so I think he had the same troubles as I.