How to install casings on 5/8" drywall and 1/2" window/door jambs
You have a few options:

Place the casing over the drywall and jamb as usual, and let it tilt a bit due to the 1/8in difference.

When you apply the caulking to the seams, the difference will not be noticeable.
The bevel joints at the jamb corners will be slightly dipped, and you can minimize the visual distraction by applying some drywall compound and sanding it.
The drywall panel and the jamb are rarely ever perfectly flush anyway.

If however you require more precise results, you can add a 1/8 filler such as strips of &quot;door skin&quot; at the jamb so that the casing fits flat over the drywall.

Then apply caulking to fill the 1/8 gap at the jamb.
You can also combine the two and fill with only 1/16in. This makes the gap smaller but leaves a bit of a tilt.

with a table saw, trim off 1/8 thickness from the bottom of the casing, along its length, but only for about 1/2 of the casing's width.

If your jamb is 1in thick, and you want the casing half-on (1/2in on jamb), and you have 1/2-in shim spacing, you'll have 1in that needs full casing thickness, and the remainder will be over drywall and will need to be trimmed. For 2in wide casing that's half its width, for 2.5in wide casing (most populat) that's 1.5in of the width that needs its thickness trimmed etc...
This will provide a snug and level fit on both the jamb and the drywall. You need to set the blad height to half the width of the casing, and the fence at 1/8 (almost at the blade) and feed the casing on its side.
Which one you choose is a matter of aesthetics, tools and time.