This set uses two tubes in the FM section that are not used for AM: the 2nd IF amp (6BA6) and the Discriminator (6AL5). If either of these tubes have bad emission you will have a dead FM band.
You can swap around the 1st and 2nd IF tubes and see if that makes a difference (it will disable both bands instead of only FM if the tube that is presently in the 2nd IF socket is inop, since both AM and FM use the 1st IF amp, but only FM uses the 2nd.)
The 6AL5 is a cheap double diode that can be easily bought on eBay or one of the online used tube vendors.
Another possibility is that one of the two FM IF stage transformers has an open winding, and can't couple the signal to the next stage. You can "bug" the FM transformers (with the set unplugged) to figure out if this is the case or not.
The voltages that you quoted are a tad high but not so much that the set wouldn't work unless **something else** is very wrong.
The resistor burning up in AM mode is a mystery that you really need to solve first. It's a fault that could have serious consequences for your safety if you blow it off- replacing the burned-up original resistor with a higher-wattage one is not a whole lot different than putting a penny in a household fuse socket. And who knows what else in the set you are overloading.
I guess that it's possible that the mixer tube can't oscillate at the FM IF frequency. There could also be a defect in the oscillator tank circuit- open coil, shorted trimmer cap, damaged tuning cap...
No way to tell what low emission would do in any given circuit, short of saying that "it won't work right". Sounds like replacing the 6BE6 tube with a known good one is the only thing that you've got left to try.