Was the motor replaced with the same size (horsepower, amperage, voltage) motor? What is the amperage and voltage listed on the motor's data plate? In many cases the motor plate will either list the size of the required breaker, or if not, the manufacturers website will have that information available. If you "had the motor changed" it might be best to call the company that changed it and report that the new motor is tripping the breaker, where the old one did not, and have them check for issues, since they should have verified that the new motor would work the same, or made any changes needed to support any change in the motor. If you changed it yourself, that's on you.

You should not replace the breaker with a larger breaker unless the new motor requires a larger breaker.

You should not replace the breaker with a larger breaker without ALSO replacing the wire with larger wire, suited to the size of the larger breaker, which should be suited to the size of the new motor. 20 amp, 12 gauge copper, minimum - 30 amp, 10 gauge copper, minimum.

Rather than "sanding the wires" you should (with the breaker off, of course) cut off the damaged ends of the wire, pull it a little further in, and strip the ends to get new undamaged wire. If this is not possible, consider replacing the wire, especially if it's fairly short run.

110V (single-pole breaker?) is rather abnormal for most well pump installations. One potential fix would be to re-wire for 220V operation if possible

To do this properly, You'll likely have to run a new circuit. Since you haven't posted the make, model, or nameplates of the equipment, it's impossible to say for sure if these two devices can be on the same circuit. If they can, the solution is to extend the circuit using approved methods and materials. If they can't, the solution is to run a new circuit.

As for making an extension cord, that's not the best option. Extension cords are for temporary use only, and should not be used in place of permanent wiring.

WARNING:

Using an extension/adapter cord like this is dangerous, and can lead to loss of property and/or life. This is a theoretical device, that should never exist. DO NOT make, use, or even think about a device like this.

If you're dead set on making and extension/adapter, you'll want to use proper sized conductors and protect the cable from physical damage.

Since you're plugging it into a 50 ampere circuit, you'll have to size the conductors to be able to carry 50 amperes. This is not typically the way it's done, but this is not a typical solution. If the A/C overloads, you don't want the cord bursting into flames. So since the overcurrent device is set to 50 amperes, you'll have to have appropriately sized conductors.

A safer option would be to build a smaller overcurrent device in to the adapter, to protect the cord and A/C unit. In this case you'd have 50 ampere conductors attached to a NEMA 14-50 plug, with the other end of the conductors connected to some form of 30 ampere overcurrent protection. Then from the 30 ampere overcurrent protection, you'd run 30 ampere conductors to a NEMA 6-15 receptacle. Then you could plug the A/C into the 6-15 receptacle.

WARNING:

Using an extension/adapter cord like this is dangerous, and can lead to loss of property and/or life. This is a theoretical device, that should never exist. DO NOT make, use, or even think about a device like this.

The safest option would be to install a new panel, and have it being fed by the conductors that are currently feeding the 14-50 receptacle. Then you can branch to each of the devices from the panel, using proper methods and materials.