I remembered that I had started taking slippery elm to sooth my stomach. When my mouth started to get dry I stopped. Could the SE have caused it?

If you had Sjogren's then this could be a possibility, but you would also have other symptoms such as dry eyes as well.

Does this mean that perhaps I have an immune issue since I still have the dry mouth?

There are various causes for just the dry mouth such as diabetes, nerve damage, various medications, age, Bell's palsy, etc.

I am going to the doctor next week, is there a test I can ask him to run?

The doctor can start with a history and some basic lab tests to see if anything pops up as a red flag.  If those are normal then the doctor could run additional tests for things like autoimmunity.

I saw on another post of yours, your recommendation for D3 for immune issues. Would it be something I should try and if so, in what dosage?  I am still working on improving my adrenals.

The max I would recommend would be 2000IU daily, but until you find out for sure what you are dealing with I would not even recommend anywhere near that amount, maybe more like 800IU daily.