If its overheating at 35 F with decent fan something is seriously out of whack.

As a useful and low cost diagnostic, suggest you remove the T stat and run it. As mentioned above, run it with the cap off for a while and watch for coolant flow in the top tank.

If removing the stat improves temps it is a bad stat. If you don't see coolant flow either the radiator is seriously blocked or the water pump is fubar. Its rare, but the blades can rust off the pump.

If you suspect the radiator, you can pull it and get it checked out at a rad shop.

Also, mild "at home" flush of the block and radiator wouldn't hurt. I rig up an adapter to go from garden hose to radiator hose. 1 1/4" or 1 1/2" PVC pipe matches rad hose diameter pretty well. With stat removed, rig it up to flush the block or radiator.