Ok, couple of things here.
First, on the governor which you apparently futzed with. There are FOUR "things" on the governor that are "adjustable" without removing the cover or dorking with internal adjustments.
The first is the speed lever. That one is obvious - it is connected to the cable that goes to your throttle control. You set the speed of the engine with this; the governor controls the fuel to result in the speed you select. It is on the SIDE of the governor.
The second is the STOP lever. That's on the TOP of the governor cover, and is pulled to stop the engine. It must be all the way to the RUN position - if it is even a tiny bit towards STOP it will screw with you badly, including producing hard starting. Make absolutely certain it is not binding and goes ALL THE WAY to RUN when not in use to stop the engine. Your problem is likely here as it sounds like you diddled the linkage to the STOP lever from your description. Undo that.
The third is the idle speed screw. This is found on the end of a small cylinder near and behind the throttle lever's shaft and is an allen screw with a locknut around it. You need a small allen key to turn it. It faces AFT. The one facing the side of the engine (usually starboard) is the buffer screw - I'll get to that one. The aft screw adjusts the idle speed ONLY, but beware as the buffer screw interacts with it - procedure follows.
The buffer screw is on the SIDE of the governor case, and is a hex bolt with a locknut around it.
To properly set the idle speed you need a phototach or other ACCURATE means of measuring RPMs. Bridge tachs are not adequate.
To set the IDLE SPEED, warm up the engine, back off the BUFFER SCREW a few turns (3-4 will do) and then loosen the locknut around the idle speed screw. Set the idle speed using a phototach and an allen wrench. 500-600 RPM is recommended for marine engines; I prefer the higher end of the range unless there's a problem with idle speed when set there (on some boats there is.) Holding the idle speed screw, tighten its locknut. This requires both an allen key and open end wrench. As you diddle this you will see that the throttle lever moves slightly - this is normal, as what you're adjusting is the "low side" travel of the throttle lever.
If idle is set too low, you will likely get an engine that fires but won't start when cold, stalls going into gear, and may ring the low oil pressure alarm when going into gear hot, as the RPMs will get too low. Very low idle RPMs are not desireable; you want 500-550 in gear, which usually requires 550-600 in neutral, again, both hot.
NOW, with that done, you must set the buffer screw. Briefly pull and release the STOP LEVER. The engine will likely "hunt" a bit before it settles down. Now turn in the buffer screw slowly until the RPM goes up JUST A TOUCH - NO MORE than 15rpm - just enough to stop the hunting (5-10rpm is usually enough.) Try the "stop lever" test again. IT IS IMPORTANT THAT YOU NOT TURN IN THE BUFFER SCREW MORE THAN 15 RPM! Then lock that one down.
Re-check your idle RPM to make sure nothing moved while you were locking it down. If it did, go back and do it again from the top.
Test going into and coming out of gear; it should smoothly do so without hunting and NOT "dawdle" at the higher "unloaded" RPM when coming out of gear.
DO NOT attempt to adjust the idle speed with the stop lever or with the buffer screw - you have to use the right tool for the right job.
Detroit governors can be flummoxed if toyed with inappropriately and in extreme cases (particularly abusing the buffer screw) you can "lock up" the governor and cause a full-rack runaway.
If that doesn't solve the problem you need someone to come out and go through the full tune-up procedure. Not a big deal, but it has to be done "by the book".