Yes you have two main bolts on the underside there, one on each side of the seat post, bolt heads facing down. Some saddle clamps have them fore and aft, some only have one, but same principle. There's basically a two-part clamp in the centre, close to the seat post- The top half on the top of the seat rails, and the bottom half under them, but with a rotating "U" shaped piece which controls the pitch of the seat. Those one or two main bolts just bring the two halves of that clamp together and secure them against the curved bit down below.
As you found, if you undo both bolts just a little, the whole saddle will suddenly feel less stable, that's the clamp released. The saddle can now be adjusted in all planes with the same bolts, that "up and down" you're feeling is the lower clamp cradle rotating (you should see a sort of "U" or cup shaped piece low down, engaging with a lower plate, sometimes on a serrated contact).
To adjust fore and aft, just slide the seat forward or back... give the back or front of the saddle a little nudge if it's sticking and it should go. It's sometimes a bit fiddly to adjust fore and aft AND keep the seat level, but you should get it okay with a bit of playing.