We have a Coleman 7300 rooftop ac unit. When the

To be absolutely certain, remove the old one, read the values and purchase a new one from an electrical store.

relay modem, capacitors, or bad compressor

I would check thermostat first. Remove compressor wire from thermostat and see if that cures starting by itself.

If this is a 'new' unit I would contact Coleman and explain the problem you are having. I would think that they would have to honor the warranty on the unit. That is, if a AC company installed the unit. However, from your explanation of the problem it appears that you might of installed the unit yourself in which case - 'they' (Coleman) might decide to 'not' honor it. A gray area for sure. Still, I would try and see if they would help you.
In the case they won't - I would get ahold of a AC Tech and have him/her troubleshoot the unit before I started removing and adding electrical components.
From what you've said - I wouldn't think you have anything seriously wrong - you just need someone to figure out why it's not working as it should.
My best guess - is that it's a simple electrical problem that can be solved with one service call and for a minimum cost.
Worst case - might be a major component (i.e. compressor) that is bad (which is not likely) but in any event with the AC Tech's help - Coleman would probably honor the warranty on the compressor and you would only be out the labor for the AC Tech to install it. Note: some warranties will even cover the labor to put it in.
Bottom line - I would try to get Coleman to help me directly or with the help of a AC Tech, and in the event they won't - I would trust the AC tech to find the problem and repair it.

Sounds like you're issue will be the Cap do you have a multi meter?

COLEMAN R.V. AIR CONDITIONER SERVICE AND REPAIR QUESTIONS
COLEMAN R.V. PRODUCTS
GO TO rvcomfort.com
Click on RVP (RV PRODUCTS)
Click on type of AC, if you can’t find the information you want there,
then click on contact us on the Green Bar.
You can get Service Support or Email address and Phone numbers.

Replace the start capacitor for the fan and compressor. Check all the wires to the capacitor.

When they sit allot - sometimes the fan will stick.. Take the cover off,, turn it on and see if the fan is stuck.. Spin it by hand to free it up..

I fixed it. I had a dirty connection on the starter relay for the compressor. Thanks again for the help.

could be bad start capacitor for compressor, cheap fix, anything else could be internal in system or compressor