Can muscles tighten and give you limited mobility if you work them out and don't stretch after?

I went rock climbing for the first time a couple of weeks ago, and definitely ended using muscles in my body that I had no idea existed. I also have a fused spine, so I'm sure those muscles in my back have atrophied and combined with my limited mobility, I had to compensate my movements.

I am noticing that the outside edge of my arms (like right by my underarms) are SUPER tight and sore STILL... and it almost seems that I don't have the normal range of motion I used to have with my arms. Even something as simple as trying to stretch by bending my arm and putting behind my head/pulling gently on my elbow, is excruciating. It's just super tight. It's an equal pain on both sides, so I'm pretty confident I didn't injure one of my arms.

Could this be caused by not stretching after climbing or does it sound like something more serious?
Can muscles tighten and give you limited mobility if you work them out and don't stretch after?
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