I'm also about 99% sure you need a new motor. The esc is sure questionable too , but even though the esc is what controls the beeping it is the motor that is doing the beeping. Perhaps the bad motor will no longer beep even with a good esc? I take it that it doesn't even spin up propless either.
I will tell you what I think happened , but I sure don't know why unless it was over propped or over volted (like using 4s when 3s was called for)
I think that your first few flights overheated the esc because of the motor/prop pulling too much current. The esc saved the motor by doing what it's supposed to do and cutting out in 2 minutes. The last flight allowed the motor to overheat for a longer period and didn't get saved by the esc.
A 4 minute flight tells me that your set-up is pulling 37-28 amps average and possibly peaking over 60 amps. I'm not sure about the plane , but that's a "hotrod" flight , not a lazy flight.
The cub should probably fly over twice that long. Was the plane very fast for a cub? It should've been.
Here's one thing you can do is to disconnect the motor wires from the esc and make sure they are'nt touching each other. Spin the motor by hand and if it's still hard to spin then it's tied up. I'm not sure how to diag the esc other than just using a different motor or esc to eliminate one component.