From your description of the issue, I see several possibilities. The first may be the amount of grass you are cutting. While mulching, which is the hardest type of mowing for a mower, as all the clippings are contained in the deck and typically cut multiple times before dispensing back into the yard. Rule of thumb has always been to cut no more than 3/4" of grass at a time, which may mean that you will either have to mow more frequently or go over your yard twice. Cutting much more than that will 'choke' your mower deck. The reason the issue is on the right side is because the left side is throwing all of it's clippings to the right side, so it is having to deal with twice as much grass.
Now, you also mentioned that you tried it without the mulch plug and it was better but not great. This would lead me to believe that you may also have a spindle bearing failing and/or the belt is worn and slipping. I would recommend visually inspecting your belt for wear and the spindles. If you feel any resistance in the spindle, it needs to be replaced.
The only other possible issues I can think of are: Are you using Mulching Blades? Standard blades create too much airflow and will actually lay the grass down before it is able to be cut, hence, the straglers. Are you running the engine at Wide-Open Throttle? Very important to do so. The last suggestion I have is to make sure the front deck skirt is not bent as that will have a drastic affect on the airflow. If it is bent up, it will cause air to blow downward, pushing the grass down and keeping the blades from being able to cut it.
Good luck!