I just had my oil changed today. The oil smelled burnt before it was changed so I got it changed and thought I was good to go. Then, I realized I was still smelling it and my oil pressure sensor light has been coming on. Sometimes when I shift, the car will vibrate and sort of rev up for a second or two and then stop. I checked around under the hood and did not see any leaks and I haven't seen any oil on the ground. I am having issues with my cooling system as well. Could the two be directly related?
This sounds like you are having three different problems:
The engine had burned oil and you changed it and now the oil light is on.
The transmission slips and then catches.
The last thing you mention the cooling system has a problem.
They could all be related to the cooling system overheating the engine and transmission. There is no way to know what is causing the three different problems without a thorough diagnostic inspection of the oil pressure light by a qualified mechanic. You did not list the vehicle, engine or year; it would be hard to identify possible causes with out this basic information.