If they are similar to the gen 1 rear brakes, the pad holder stud/nuts have cone washers. You loosen the stud nut which allows positioning the pad along the rim. This also allows adjusting the heel/toe.

My understanding is that the gen 2 added the contact adjustment screws you mentioned, for gen 1 you actually have to add/remove washers from behind the pad holders to adjust the contact point.

Take the rear wheel off, remove the speed fin, press the brake arms together by hand, and slide the brake wedge out, and leave it dangling. Now you see two M4 hex head bolts that hold the brake assembly to the frame. Loosen these two bolts, and adjust the bracket, so the brake sits perfectly centered.
While you are at it, check if both brass barrels that actuate the brake, spin freely. If not drip oil on them, and try to free them. (You can gently use needle nose pliers). They should spin easily!
Now loosen the brake pad holders, check if both cone washers are present, and ever so slightly tighten them back up. Put rear wheel back in, leave wedge out. Check alignment of pads. And pre adjust. Put wedge back in. Use the worm screws on both arms to tighten brake up, loosen brake pad bolts, adjust pads so they sit where you need them to be. Tighten brake pad holder bolts. Loosen worm screws evenly till you got good brake engagement, and all is centered.

Or … take it to the shop.

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I love rim brake bikes!

Thank you! I'll definitely give this a shot!