Can I submit insurance forms myself?

I have done this. You will just have to pay upfront. Make sure the bill is marked paid with your check no. or cash and you may need a diagnosis number. Ask about that. Make sure the doctors name, phone and address are on the bill. I had a bill that the stub for payment was the only place the doctors name and address were. Writing it in was not enough so the insurance company wouldn't reimburse unless I submitted a bill with name and address. For $10 I wasn't going thru that trouble.

I have done that. The forms are available online and the people on the 800 number were very helpful. Your counselor will probably have to give them some information about the treatment you receive but you should be able to file a claim yourself.
If your counselor will not even provide information to the insurance, you might want to find one that will.

You should have a membership card for your insurance. On the back of that card should be a toll-free customer service number. Call that number and they should be able to help you.