Crossed batterie cables now no start

The solenoid has nothing to do with it starting. It only allows the starter to work and you said the engine was cranking. I would check ALL OF YOUR FUSES. There are some inside of the cab and some under the hood. If all of the fuses are good I would hook up a scanner and see if it has any codes in it. It's a possibility that the ecm/pcm was damaged from hooking the battery wrong. It could also have damaged the ignition module. But lets start with the fuses first and the scanner. You can get a cheap on at autozone and you need to get one that will work with your vehicle. It's not an obd II vehicle and it uses the ford specific connector which is located under the hood on the firewall usually.

Your description of the solenoid replacement is not clear. If your original solenoid had two small terminals and the replacement has one, then that is possibly where your problem may be.