A few questions..
What engine do you have?
What year?
Is it a carbed or fuel injected engine?
Does the oil pressure drop first then the engine die or does it all happen at once?
Also, what is your oil pressure? (assuming you have an actual guage)
I would actually start with something a lot simpler, say a full tune-up, including carb adjustments as needed.
See how it performs from there and decide if further action may be required or not.
I just wouldn't go into ripping apart the engine in search of something it isn't first.
Now, as for the overheating question..
Does it overheat or get what you would consider too warm? (190f +)
Even with no visible leaks, do you have to top off the overflow or radiator often, such as every few days?
If no to those, then a flush may not be needed, but certainly won't hurt but may not have anything at all to do with the stumbling. Unless you have oil in your water and vice versa. Then it would be time for an overhaul or a new engine.
All in all, it really sounds like a simple issue that can be corrected easiy with a good complete tune-up including fuel filters and a fuel pressure test.