First I am glad to hear that he can use the leg and injury is unlikely. Now based on his signs, the weather, and at his age, arthritis could be what is getting him down. We often see this cause issue for elderly dogs this time of year and it can leave them quite stiff and sore.
In this case, there are some things we can do to ease his signs and help with the winter ahead. To start, we do find that supplements like glucosamine, chondroitin, and omega 3+6 fish oils helpful. You can get these OTC at most pet stores or even use human preparations of these to help him keep mobile.
As well warming the joints, can help fight the effects of the chill. Over the winter he'd benefit from a bed in a warm spot, access to a heating pad or even a heated dog bed. As well for the moment, consider warm compressing this sore hip.
Finally, while we'd want to think about speaking to his vet about dog safe pain relief for him, for the moment you can try him with a low dose of buffered aspirin (5mg per pound every 12-24 hours). Of course, if he is on any other medicine or has any known health issues, we'd want to check with his vet before use.
Overall at his age, this is likely arthritis related, so we can use the above for now and as needed to keep him comfortable and moving through the chilly weather.
All the best,