If I'm getting water in my transmission from the cooler lines, can I get trans fluid into the regular coolant in the radiator?

 

There is a heat exchanger inside of the radiator for cooling the transmission fluid. If you are getting water in the transmission it means the heat exchanger is leaking. So yes, you can have transmission fluid leaking into the radiator and radiator coolant getting into the transmission. Not a good thing.

 

Can I plug the connections on the radiator and just use an aftermarket trans cooler?

 

Yes you can, but I'd be concerned that may be a temp fix. If the cooler in the radiator is leaking, then the rest the radiator is probably close to going too. I'd also be very concerned about getting the trans completely cleaned, and that may take removing it to get the coolant out of the trans and converter.
If you can get the car to a transmission shop after rerouting the lines to a remote cooler, they can backflush the system and probably get it cleaned out, but don't forget to change the filter too! Get it done soon before the water starts to damage anything!