Not too far from the campground entrance is a trail that will take you over to the North trailhead. It's not too far

If you request the stables campground, imho the best in the park, you can walk the equestrian trail to the mountain bike trail that goes around the lake. Also more connecters to other mountain bike trails. Just be mindful of bikers, clear the trail when they need to ride through. During the week you will encounter few cyclists or equestrians. Just keep in mind the stables campground is best for sure.

My recollection is the trailheads are not that close to camping. Definitely check the maps before deciding.

I remember a couple little trails connected directly but the main trailheads are far to each end of the park. The good news is the end where the campground is, the trailheads are just a little ways downhill past the campground entrance. Our site was about as far from the entrance as you could be and it was right at a mile to the main trailheads. You might also consider walking the edge of the lake. It's not a marked trail but it is trodden. Here's a link to the map too: https://www.alapark.com/sites/default/files/2019-07/OMSPparkmap_0.pdf Happy Camping!

They are a pretty good distance by foot (maybe a mile to the closest one) so we drove to them. The campground is large enough to go for a nice walk though, and you can walk to the lake at least.