On the older bikes that have those adjuster screws at the rear drop outs, how do you know if your wheel is square? Do you trust the drop outs and evenly adjust those adjuster the screws?

Look for equal gaps between the tire and chain stays when pulled back against the stops. Doesn’t take much of a tweak to those dropout screws to change that alignment.

Back off one adjuster screw. Adjust the other screw to position the axle where you want it in the dropout. While pushing or pulling the wheel back against the adjusted screw, centre the tyre between the chainstays and lock the QR skewer. Now tighten the backed off adjuster screw against the axle. The next time you install the wheel, all you should have to do is pull it back against the adjuster screws and then tyre should be centred between then chainstays. Of course, this assumes the wheel is properly dished and there are no frame alignment issues.

I would think if you installed a different rear wheel and it remained centered between the stays it would give you some degree of confidence that the wheels' dish was correct. There are ways to check for alignment of the frame itself using improvised tools (or a shop can do that).

My wheels aren't perfectly square, but I don't think it affects the ride that much.

A long time ago I saw an image of the Lotus F-1 in Autoweek when it was published on newsprint... The team using these wheel replacements, all metal, a hoop of steel as the rim/ tire, dished to place the outside edge at the centerline of the actual rubber tire...
For alignment and suspension configuration...
Made me think about bikes... So, mounted true, dished wheels without tires on my bike and while in the stand took a straight 1x2" hollow extrusion and aligned it against both rims as close to center as I could go without spoke interference, made the dropout adjustments from that.
I was pretty close but the bike rode "no hands" easier after that session.