Ok, Then here's my opinion of what's happening:
These carbs are VERY sensitive to scale deposits in both the bottom of the float chamber and in the accelerator pump passage and orifice. Not draining the carbs regularly leads to the sort of "runability" problem you are describing. I actually believe that running the engine until it quits may even exacerbate the problem since it doesn't remove the fuel from the bottom of the chamber but depletes enough liquid in there so that continuing evaporation and scale deposit is accelerated.
The only way to get your baby back on track is doing a thorough cleaning of the carb with QC (quality control) checks as you go. Since you already had it off once, it should be easy to take it off again and get it straightened out. I do suggest that you purchase a few critical parts before beginning but, sometimes, the task can be done reusing just about everything. Here are the parts I recommend having on hand:
Item # 1 gasket set..... X 1
Item # 19 gasket........ X 2
Item # 4 screw set.... X 1...This is the idle mix screw and can be easily broken during removal. If you are careful and don't break your old one, then you might not need this.
Refer to parts link illustration below.
http://www.boats.net/parts/search/H...6/BFP9.9D6 LRTA /CARBURETOR (AUTO)/parts.html
You may also find that your "jet set" (item #18) is cracked or has debris inside of it that is impossible to remove. In that case, you would want to replace it. You won't know until you get the carburetor apart and inspect with a magnifying glass. I replace them every time I do a carb just because but you may not want to spend the money. Your choice.
Notice that I used the example of an auto choke carb. If yours is a manual choke (pull start only) you will still need the same parts but you should order from the page that best describes your model. There are substantial savings in ordering from the boats.net site as opposed to going to your local dealer. If you want to proceed without replacement parts, ok but you may not get the best results.
There is also a carburetor repair manual for Honda outboards available from Helm Inc. I have never seen one but hondadude says they are quite good and I trust his opinion. If you just want to wing it with me to get it cleaned up, then that's ok too.
One more question: does your accelerator pump (item# 14) actuator pin move down and then back up when the throttle is moved from idle to wide open then back to idle? This pin is the one inserted through the little rubber bellows or "cap" (item #12) on the side of the carb.