Onan 5500 Runs then shuts down after 10 min...?

Is it giving you a code after shutdown? The start button will flash on and off a certain number of times after it shuts down. The number of times it flashes determines which code it is giving you. You should be able to look up the code number in the Onan manual or online. That code normally (not always) indicates what is wrong.

Check oil level. It has an auto shut-down on low oil.

If it runs fine and suddenly shuts down, this definitely sounds like a sensor is shutting it down. Now is the sensor bad, which sensor is it, or is the sensor doing it's job and there is another issue?

I wouldn't waste time and money for parts that may not be needed. I would take it to an Onan shop and have it fixed right the first time. Of course I full time and I want my Onan ready to work at all times.

Sounds like it needs to be diagnosed before replacing parts. Not that replacing a part is always a bad thing in comparison to labor rates vs chance of fixing it. It takes 3 basic things for an engine to run. Compression, fuel, ignition. About the only way it would lose compression is for a valve to be hanging up when it heats up. This can happen with a build up of carbon on the valve stem. It's probably one of the most unlikely things to happen but....

You may be able to narrow it down to fuel or ignition at the moment it cuts out. If you were to hold a spark tester near the spark plug and watch for the tester to stop flashing that may tell you if you are losing spark. To further isolate the problem on the ignition side you would need to move to testing why the spark is dropping out. Systems that could cut the spark out are low oil pressure or over heating. Those 2 protective circuits are utilized on many small engines. Be sure the oil is up and the cooling air system is not blocked. Mice nests, plastic bags etc.

Sometimes oil pressure shut down switches will leak internally and shut down. Need to trouble shoot the shut down system.

Another thing that could be done at the moment it dies out is to supply a secondary method of delivering fuel. One way might be to start manually closing the choke if so equipped. The other way would be to fake the fuel supply and give it fuel by either very slightly supplying some starting fluid or gasoline to the carb intake or air filter intake. I DO NOT recommend using starting fluid as it's hard on the piston rings as it's very flammable and violent. I prefer to use a small spray bottle with gasoline. ANY method of enriching such as this can be dangerous. I dont know your abilities so I'm suggesting that if you are not confident in doing this type of troubleshooting take in to be repaired.

Any time you work on something like this or any small engine move it outdoors and have a fire extinguisher on hand. Again, you are working with fuel and ignition sources so it is dangerous. Keep the amount of fuel used in testing to the bare minimum. Watch for any spills and dont check spark or ignition/electrical with fuel spills present.

If you supplement the fuel system at the moment it starts to die and it then recovers and continues to run it is telling you that you have lost your fuel supply. You then need to troubleshoot the fuel system. If it still dies out when supplying fuel it may mean that you have lost ignition.

One example of spark tester

This may be close to your model
Need exact model number

sounds like an overheat issue to me

Had the same issue with our Onan Microlite 4K. Turned out to be the fuel pump. Ours is gas fuel and has a "prime" function on the start button at the generator. If your has the same, try pressing prime after the gen shuts off and listen for a pulsing sound from the pump. If no sound immediately after the shutdown, but sound after waiting 30 minutes, replace the pump.

could be a bad coil ,they crack inside, heat up and then short out.