Well, it does sound like a fuel issue but let me ask you a couple of questions.
Did the engine run great before you put it up without draining the carb? It in no way exhibited this symptom before you put it up with gas still in it?
If not, then I highly suspect that the new carb didn't get set up properly at Honda. Either that or it's possible that it isn't precisely the correct carb for your outboard. Before doing anything to the carb, you may simply want to exchange it if you can.
BUT...
If it was stored long term in a location where bugs or rodents could "set up shop" then it's possible that something critters did or left behind is, in some way, causing you headaches now.
Wasps are notorious for leaving mud "dobbs" inside crankcase vents, underneath flywheels and various other hard to see places.
And we all know how rodents love to chew wire insulation.
Getting back to the carb though....
The idle circuit does contribute fuel during high speed operation so, if it's too lean, it could cause some loss of performance.
However, the surging could be because the float height adjustment wasn't done correctly. Or, the inlet valve is sticking for some reason. I have flooded the float chamber with WD-40 by draining the fuel and, with the drain screw still open, sticking the spray straw in the drain and blasting away. WD-40 won't harm your engine UNLESS it pools up in the intake and hydro-locks the cylinder. So, when "flushing" like this, pull the spark plug and spin the engine a bit for safety's sake. Sometimes this flush trick works. Most times not.
You will have to take the float chamber off to check/adjust the float setting if you suspect that is the problem.
Yes, the limiter cap is "glued" to the head of the idle mixture screw with Loc-Tite. It can be easily removed by applying heat from a soldering iron. The cap is usually destroyed while doing removal this way. You can purchase a new one though if you want it on there. You're "technically" not supposed to do this but I have yet to see the Limiter Police come for anyone.
I hate to tell you that I've purchased carburetor float chambers directly from Honda Genuine Parts when I worked for a dealer that had sand from the casting mold still in them. It is rare but it happens.
There are numbers and letters on the carburetor that can help identify them. Make sure that the ID on your new carb matches your old one. If not, you'll need to find out why.
Double check the correct part number for the replacement carb according to frame, serial numbers and year model. Verify that is the part number that you were shipped.
Can't think of anything else at the moment but I'll get back to you if I think of something else that you can try.
Good luck.