Sounds like the oil lamp is triggering the ESG, causing rough running and speed limiting to under 3,000 rpm until the problem is corrected. Restarting clears the error. That's by design. You may have low oil pressure, or a false detection of low oil pressure.
If your oil is even a little over full, this could happen due to aeration of the oil. Likewise, if the oil is diluted with blow-by products, it will be thin, and the oil pressure will be a bit low -- really low when you slam the throttle shut, triggering a low oil pressure condition. We usually fill the crankcase to half way on the dipstick. Of course, you could possibly have a bad/marginal/heat-sensitive oil sender, or an actual oil pressure problem.
I would start with an oil change. Be precise when filling -- not even a drop over full -- a pint and a half will do it. Half way on the dipstick is fine. Then try it out again. If that doesn't fix it, you will want to check the oil pressure system. If you put a gauge on where the oil sender screws in, you should get over 10 psi at idle, and about 40 psi when throttled up. If your pressures are correct, you probably have a bad sender.