Is it permissible for me, as a person who is sick and unable to move easily after waking up, to pray Fajr before or after I sleep, knowing that I need medication for sleep and another for movement?

It is obligatory for you to perform the Fajr prayer on time, to the best of your ability. If you wake up, wait until you are able to move and perform the prayer with all its conditions and pillars. If you fear that the time will elapse, then pray while sitting. If you are unable to do so, then pray on your side. If you are still unable, then pray while lying down, gesturing with your head, making your prostration lower than your bowing. It is not valid to pray it before its time nor after it. We advise you to turn to Allah Almighty and seek a cure for your illness.