I'm not sure this is a rigidity problem, but some men do have certain things that can be just the way you are or could be a sign of a problem. Only seeing a specialist would probably tell you that.
So, is this something new that is happening? Have you noticed it before or has something changed that this feeling is there. There is something called 'hard flaccid syndrome'. https://www.urologynews.uk.com/features/synopsis/post/hard-flaccid-syndrome It's a little off but does have some symptoms you describe. Do you have pain upon ejaculation?