Because I have an oil leak I suspect that the lower engine mounts have disintegrated to some extent, causing some engine vibration under load. The vibration disappears once the transmission is placed in neutral or park. There is no engine check light on, so I do not suspect any engine misfire issue or ignition problem.
Oil leakage would tend to disintegrate the passenger side one first, it's easy to check it's just inside the wheel well right behind and below the a/c compressor and behind but in front of the brake assembly. Come to think of it the driver's side transmission mount isn't much further off, sits right behind the brake assembly I believe, just on top of the frame and both of the mounts are just in front of the CV axle.
Definitely would check those, one easy way to do it is to open the hood and get in the car... Start it and apply the brakes and keep the brake pedal depressed or even apply the parking brake. Now watch the top of the engine cover from inside the car and put it in drive, wait until it engages. Once engaged put it in reverse and again wait for it to engage. Once engaged if you still haven't seen anything go back to Drive with the selector.
What you are looking for is a sudden move of the entire engine block.
If a mount (or two) are broken the engine will move considerably and suddenly (it's like a WHOA thing, you can't miss it) when going between D and R or R and D...
If the mounts are whole the engine should barely nudge a little.
That or you can visually inspect them, for this I would recommend jacking up the car and pulling the wheel off each side, you might be able to get a good look also by just turning the steering wheel all the way to one end or the other.
I would do this soon, you're not doing yourself any favors driving around with broken mounts, you will also need to take care of that oil leak... Usually it's either the oil drain plug gasket or seal gone bad (yes these can go bad in as little as 2-3 oil changes, I've learned to replace this rubber gasket on the drain plug each oil change). Another oil leak location would be the oil pressure sending unit (or o-ring), the oil filter adapter gasket (the adapter is also the housing for the oil pressure sending unit), the front and/or rear valve cover gaskets... And last but not least it could be the oil pan gasket (this is a royal one so I'd save it for last).
Generally speaking the order of replacement here is oil drain plug seal or gasket first, then move on to the other stuff (usually the oil pressure sending unit and oil filter adapter gasket are next), the front valve cover gasket isn't too bad but the rear one takes an afternoon and requires replacing the coolant crossover pipe gasket as well).
Images 1 and 2: Motor Mount, passenger side before / after
Images 3 and 4: Trans Mount, Driver side