Mine was my left hip and I had exactly the same sleeping. In the end I had to totally stop sleeping on that side. That didn’t cure it but it made it less aggravated. Try something like a big ‘pregnancy pillow’ that you can position to ensure the best sleeping position.
Another option, which someone suggested to me, is to speak to a good sports physio. They can help identify the source of pain and make it easier to work out what it is. I’m not sporty at all, by the way. It’s just that they have experience of pinpointing injuries. Might be an option while you’re waiting for medical input.
Also, could it be some form of sciatica? A nerve getting trapped?
It was actually me that identified mine was a torn ligament. I found the GP didn’t have much experience and the local physio didn’t either as she mainly dealt with retirees and age-related problems.
One caution - make sure you don’t subconsciously adopt an awkward position to protect your hip. I did this and caused additional problems by that strange posture. I found very gently, very limited stretches helped a little as mine was prone to stiffness too, especially in the morning. Check your seats, your cushions, your driving position in case you can help there. Certain seats aggravated mine.
I hope you get it sorted. It’s miserable to be in pain.