It is completely normal to have the two sides of your body be different. Day to day this may be radically different as well! We sit a lot at our jobs, so our hips tend to be tight. I think one important thing to remember (especially since this is yoga) is to be present and honor your body where it is at at this moment. There is no shame in using props to save your knees. If it's more comfortable to do that with your right side, do it!
In the last 25 years, I cannot tell you the number of times I've injured myself by pushing to the edge... It's been a very hard process to learn to be non-judgemental about being super tight some days.
Pigeon pose, lizard and dragon are all great hip openers, but go slowly!
The more you practice yoga, the more open your hips will become!!
You've got this!