So starter motor is spinning the engine?
If so do 50/50 test
Spray fuel into the engine, gasoline, quick starter(ether), carb cleaner, brake cleaner, ect............
Try to start
If it starts and dies, then you have a fuel system issue
If it doesn't start/fire its a spark issue
50/50 no guessing
EDIT:
reread post
Starter relay on inner fender can have 2 smaller terminals, one with the "S" is for activating it to start engine
Other thing needed to activate it is a GROUND, metal base of relay must have a good ground to inner fender to work
It should "click" when 12v is applied to "S" terminal, if not its not grounded or new relay is bad
If you jump 12v to "S" post relay should "click" closed and starter motor should activate
If that happens then problem of no Key Start will be in the red/blue wire hooked to "S" post
Could be Neutral switch preventing the 12v from ignition switch from getting to "S" post
"Starter solenoid" is an older term for starter relay
And just as a heads up
Original issue of jump starting and melted starter relay could be the positive battery cable to relay is corroded or cable from relay to starter motor is, or Starter motor itself
Starter motors require about 60-75amps to turn the engine over, thats why those cables and the main Ground cable are so large
When there are bad connections or corrosion inside cables they have to draw more amps and HEAT UP
Starter motors can do the same at end of life
So melted starter relay could be a sign of another issue
And maybe that's why you needed the jump start in the first place, your battery was fine, just not enough AMPs for the bad connections, double batteries doubled the AMPs and ALSO the HEAT that could be generated with twice the available AMPs
These types of starter relays are usually rated for 80-100amps, above that and their internal contacts can be "welded" together, which is what you described, "starter solenoid stuck on"