GFCI tripping for separate circuit
Crossed neutrals can be fixed depending on where they are crossed.

Start in your main panel. You will need to open the panel. WARNING, there is lethal voltage in there, if you aren't sure of what you are doing in the panel, hire a professional.

Look at both the breakers in question. Follow the hot wire from each breaker and find its corresponding neutral. Where the hot leaves the panel a neutral should enter. Does each breaker have its own neutral? If yes we can move on. If no, you have a common neutral and you should call a professional to fix it or start a new question because you have something that needs to be fixed.

Since we have two neutrals we now need to figure out where they got tied together. You will need to go to every box that either of these circuits passes thru one at a time.

Shut off the circuit whose outlets are closest to your breaker panel (call it circuit 1). Go to each box, you will need to open it up and check for voltage on all the hots going in and out of the box. If you find a hot circuit, confirm that it is circuit 2 by turning it off and checking again. Once we find the hot circuit look at the neutrals in the box they are probably all tied together. That is your problem. 

With both circuits off, you will need to figure out which neutrals go to which circuit, and separate them. To be clear when you are done, all neutrals need to have a path back to your main panel.

Good Luck