What is the ruling on the prayer of someone who had a wet dream before Fajr and woke up to perform ghusl after the adhan, but was delayed in performing ghusl and prayer, such that the time of sunrise arrived and he had not yet prayed, knowing that he suffers from obsessive-compulsive disorder (OCD) which delays his ghusl and prayer? And is his prayer valid if he intends it as an اداء (on-time performance) and not a قضاء (make-up), and is it necessary to specify the intention of اداء or قضاء?

You must struggle against obsessive thoughts (waswas) and get rid of them by ignoring and turning away from them. There is no sin upon you for what happened if you were unaware of the time expiring. However, it is not permissible to delay prayer beyond its prescribed time by yielding to obsessive thoughts. Your prayer, which you performed with the intention of performance (ada'), is valid, and you are not required to make it up (qada'), because the intention of performance suffices for the intention of make-up, and vice versa. The intention of performance or make-up is not a condition for prayer; rather, the intention of the specific prayer is what matters.