[skyscraper]

Alright, looks good.

For the RF, you shouldn't get continuity from 1 to 4 based on my coil.. I have the primary, which is the small coil that goes bad, marked 1.68 ohms on the schematic, between pins 1 and 3. Again, that's 13 turns clockwise, with the bottom of the coil (nearest the lugs) connected to 3, top to 1.
The secondary is something like 100 turns, connected between lug 5 and 2, with lug 4 going to a center tap to select the police band. I don't have more specific notes on it.
I would clean off the old coil (make your own notes, just to be sure. Drag a needle across the windings to count them). Once you do that check the measurements again, it's possible they are shorted, but not likely. Our two radios could also have different parts, I suppose. You aren't talking about the antenna coil, are you? That one also goes bad, from lightning.

For the oscillator coil, the culprit is the tickler winding, on the bottom of the outside coil. Over a piece of decaying celluloid. It is between lugs 1 and 2 on your diagram, 1 at the top, 2 at the bottom. 27 turns. FYI, it is erroneously marked 5.25 ohms on the schematic. Expect about 1 ohm from your new coil.
The primary is the small form inner coil, connected at the top to the loose wire, and the bottom to lug 4.
The secondary is the outer coil under the tickler, lug 5 is the top, 3 is the bottom. All are also wound clockwise looking at the bottom.

I was dreading the rewind, but it went really smoothly. The hardest part is getting them out around the capacitors. Let me know if you need any more info, everything is still on my 'work bench'