You get much better stability when you inflate your tires less. I haven't ridden real single track in a long time, but sometimes I'll ride my mountain bike on a bridle path next to a rail trail in this area. I've found that the difference between max. psi and putting less air in the tires is huge. Unfortunately I don't remember the details because it's been a while, but I think I put 30-35 psi and found I was much more stable and bounced around much less. My bike is a 26" hardtail.

You might want to see if your local mountain bike group offers beginner classes or if they have any group rides for beginners. Or check with your LBS to see if they know of anything.