Yikes, I think you may have damaged your boat. Also, you posted this in the PWC section not the boat section so you may want to repose in the correct section to get the most help.
Running it on the hose for 20 min. is WAY to long as you suspected. You are only supposed to run these machines out of water on the hose for max. 2-3 min, pretty sure it is right there in the owners manual. Reason is that the propulsion system, when out of water, is not lubricated or cooled and it will heat up quickly and wear out in no time. Take a look at your jet pump and look at the wear ring, it might be melted and/or your drive shaft seal (carbon ring) is shot. Your entire propulsion system needs to be inspected by someone with experience. You could also post some pictures here and some of use could probably indicate to you what the likely problem is. Did any of the dash alarms sound or any engine codes pop up? Running the boat that long out of water could also overheat the engine b/c the heat exchanger is out of water.
To have the dealer replace the carbon ring is probably about $500, because there is a lot of labor involved. Parts alone are probably only $150. I suspect that you may also need a new wear ring and maybe even impeller housing if it got hot and warped. It is plastic after all. Throw all that together an you are looking at over $1000 for the work.
Back to your original problem, before you ran it on the hose, the engine vibration was probably just spark plugs that needed to be changed. These engines are very sensitive with the spark plugs for some reason. Tie that together with old gas, burning off fogging oil at first start up of year, and sitting for several months and sometimes the engine just won't run well. Even though the maintenance manual does not require it, most of us replace the spark plugs at the beginning of the season regardless of hours to avoid any running problems. Plugs are cheap and easy to replace at about $10.
Sorry to hear about your troubles and hopefully the woes are not as bad as I described, but I fear the worse. An expensive lesson to learn. Good luck, and let us know if we can provide any additional info.