What you describe sounds like your unloader valve is not opening. The unloader can be set to open at too high a pressure or it's stuck closed because of mineral buildup. I see that all the time. When you are starting your unit without pulling the wand trigger, the pump must overcome the unloader so that's why it's less of a load on the engine to pull the trigger instead. Keep in mind that the pump is pumping water 100% of the time the engine is turning. It's either coming out the wand when the trigger is pulled or it's recirculating in the pump because the unloader valve is open. When you pull the trigger, the unloader closes and water goes out the wand. If you leave it run too long without pulling the trigger, the recirculating water heats up until the thermal relief opens and you see water come out of it. That's why you should start it with the wand trigger pulled. This takes the unloader out of the picture while starting and is less of a load on the engine. If it starts, it is likely that when you let go of the trigger, the engine will stall. This also points to the unloader.
Also, you said you have water in the oil. This may sound like a dumb question, but which oil are you referring to, the engine or the pump. It's highly usual to ever check the oil in the pump on the typical residential model washer. That usually lasts for the life of the unit. [I'm assuming your machine is a residential type, with a vertical shaft engine like a Honda GCV160 or equivalent sized B&S] Your machine likely has an Annovi Reverberi pump and on some models, they have a black fill plug on the side. Others require a complete teardown to change. Is that what you examined? If you have water in the pump oil, I'd definitely replace the whole pump!
A replacement pump runs about $90. You can replace just the 'top end' or head of the pump for about $60. (not an option if your pump bottom end has water in the oil) A new unloader valve is around $30. [these are prices for the typical A-R wobble plate piston pumps on vertical shaft models]
Good luck!