The answer to your question is YES! And, the other symptoms you described indicate your rudder bearings are in need of attention and/or replacement.
This is how I know this: I have a J/36 built in 1981...a wonderful boat by the way. Several years ago, I noticed water in the bilge when the boat was at the dock and only when the winds were rough and producing waves hitting the stern. I also noticed water in the bilge when I took the boat out, but thought this was from the stuffing box...but it was really more than the stuffing box should have produced. It took me forever to finally realize where the water was coming from. My bottom bearing sits just above the waterline when the boat is docked. Although I never experienced any steering symptoms, I had the sinking feeling (sorry about the pun) something was wrong. I had the yard check the rudder and steering system at the next haulout which was two years ago. The rudder was slightly loose...maybe 1/2 inch of play total when measured at the bottom. My understanding is that J-Boats have a history of rudder bearings going bad. Anyway, that haulout was just over $6,000 and included replacing and glassing in new rudder bearings. I haven't had a drop come from the stern since.
What I'd do in this situation: Contact J-Boats and have them send you the specs. on the rudder bearings and rudder post. Having this information will save you and the boatyard a lot of problems.
It is my understanding no J-Boat has lost steering ability as a result of rudder bearing problems. However, I wouldn't want to be the one to destroy the statistics.