Can I go to the emergency room to get checked out if my doctor is booked for months or will they turn me away cause Im not on my last breath?

You can't get a physical, but you can see them for any other illnesses. Tell them your problem, and say, "I don't know if it is related, but my chest and left shoulder hurts too."

There are all kinds of reasons including Covid-19 that would cause a person to be easily out of breath, so I doubt that the hospital would just turn you away.  But keep in mind, that some insurance companies do not cover non life threatening visits to the hospital E.R.  

No, please don't use the ER (or something like urgent care) for something that isn't an emergency.  You need to get tests scheduled through your doctor; the ER isn't the place for that.  They need to be dealing with genuine emergencies.

If your doctor cannot see you for months, you need to find a new doctor.  

You can go.

They just will refuse to see you, as you will be LACKING anything that actually fits the definition of -emergency-.