Steel frame check/repair
 

I purchased an old Schwinn road bike frame for next to nothing but it appears the rear is not exactly straight. When I install my 10sp Shimano wheel (it requires about 6mm of spreading to fit), the tire is completely centered around the brake mount but will nearly rub the non-drive chainstay toward the bottom bracket.

Where in Southern California can I get this checked out and possibly straightened? Even better, would there be a way to check and do this myself? Just tweak until the wheel is straight?

Given the age of the bike, I suspect it has horizontal (near horizontal slots a bit over 1" long and open to the front) dropouts. These are not intended to precisely locate the wheel between the chain stays.

Instead you manually center the rim, using the slot as room for adjustment. There are 2 basic versions , on some the right side has limited adjustment, and the axle is fully back on the right and adjusted from the left, on others the full slot is available on both sides, and you pick a central position of your choice.

that would explain why the dropouts appear to have marks in odd places along the dropouts. Thanks for the info.

That amount of spreading, the dropouts will sort of flare out towards the rear. When you clamp the wheel in, the dropouts will try to bend the axle. Exactly how bad this is is hard to say, (you might get away with ignoring it) but it's certainly not "good".
If you intend to keep using wheels with modern axle spacing I suggest you have the the frame cold set and the dropouts aligned. Cold setting is a bit daunting, but actually not that difficult.
Aligning dropouts is doable with a large wrench and a good eye, but is much easier with the appropriate tools, which a well-stocked lbs should have.

I'd slide the wheel all the way back in the dropouts and note where it sits between the stays. Then flip the wheel around and see if it sits in the same spot. If not, you'll need to get dish right before continuing... it's actually one of the easier things you can do with a spoke wrench, so don't be intimidated.

Once you have your rim centred properly between the locknuts, you can use your wheel to determine how straight your frame is.

But sometimes it's less straightforward...