After changing the hydraulic fluid and filters + screen the last time did the fluid come out nice and clean. The fluid is clean now but still jumping.
Next step would be to check hydraulic pressure, if the fluid is clean, to see if the pump is functioning properly. How would I go about checking the hydraulic pressure?
If this is true then both systems jumping would point towards air leak in pump inlet line or some restriction in the inlet line. I do I check for air leaks?
I was just looking at some info I have of those Yanmar/JDs. Boy is that a nice tractor! Most were built after 2000 so they are recent enough, but they were made by Yanmar to a design that dates back to the 1970s and 80s. Basic diesel engine and H/L 8 speed transmission. A tractor doesn't get much simpler than that, although they did try to complicate it by putting too many control levers on the dash. Ergonomics over function......
One change that Deere made could be the source of your "hydraulic jumping" problem. Deere put their own loader control valve on some of the 790s and they were trying out a new system of regenerative loader valving. It was an option, and if you have that option the regenerative functioncould cause the jumping. I'm not sure how you can identify if you have that loader valve. On the 770s you could tell because it had a longish lever for the loader that disappears into the bodywork down by your right knee. Maybe you should post a photo of that area where the loader control valve is located.
To your questions - you can buy a hydraulic pressure gauge test kit for a couple hundred bucks and generally it will have the right adapters to fit between the loader control valve and a loader hose so that you can watch the pressure guage as you work the loader. That may or may not tell you anything.
I doubt it is a restriction. More probably it is either an intake side air leak and a bubble going through... or that one of the automatic diverter valves in the system has failed. None of this is expensive, but can be hard to find. You have a diverter valve for the power steering - if it has PS -, plus several for the regenerative function on the loader control valve - again, if it has that option. And finally, the safety pressure relief valves are a type of diverter valve and could be bad.
All of these proportional and diverter valves are simply a poppet valve where hydraulic pressure pushes an valve against a spring to open an alternative flow route. In the case of the relief valves they do that to release too much pressure. But the point is they all have springs and a ball or poppet valve and springs can break and valves can stick... especially with water and goo in the hydraulic system.
You want to fix it because if it is an air leak then the hydraulic pump cavitates each time it tries to compress an air bubble and that reduces the hydraulic pump life.
Right now about all I can say is post some pictures, look for a leak when it isn't running, get a shop manual, and see if you can get some more diagnostic info - how it works with various loads, steering, using the 3pt hitch etc.
Good luck,
rScotty