The carb is still not clean or your idle mixture screw is not set correctly. It should be 2 3/4 turns out for most of the 15's. If your serial number is 1300001 or later, it should be 3 turns out.
If that does not fix it, then....Your idle passage is the problem.
I do not know what year your motor is, so I just picked one as an example. Part #11 (jet set)
http://www.boats.net/parts/search/Honda/Outboard Engine/2001/BF15A1 SA/CARBURETOR/parts.html
picks up the fuel and takes it to the top of the carburetor. Therefore, it should be perfectly clean. The passage through it is very very small. They are typically hard to clean very well, but you could try with a very small wire and cleaner.
The o ring at the top of the tube (jet set) needs to be in good condition. And it needs to seal well. So, it would probably be worthwhile to get an o ring set #1. Put a little oil or lubricant on the o ring and the jet set should pop right into place.
Once that is clean, then you need to take some brakekleen with a straw on the can and stick it into the hole at the top of the carburetor, where the jet set would go.
Look into the end of the carburetor that is toward the intake manifold. At the top you will find a pin hole or two. If you open the butterfly, you should see a few more pin holes.
When you shoot the solvent through the straw into the jet set hole at the top of the carburetor, you should see a good spray out of
all of those holes. If not, you need to backprobe and spray back through those little pinholes and keep working on it until there is a good spray out of them.
At the same time, you should see solvent coming out of a small hole at the front side of the carburetor. That is the idle air jet. You can spray back through that also, to be sure that everything is dislodged.
If you can remove the idle mixture screw, you should also shoot through that hole. Leave the screw in, at least some, for shooting fluid through the jet set hole and the air jet. It just allows more pressure at those small pin holes.
Finishing up....what is happening at idle....
The fuel goes up the jet set tube to the top of the carb and it combined with air, from the idle air jet. The idle mixture screw controls how much of a flow that will come out of the pin holes. At idle...with the butterfly closed.....the single pin hole feeds fuel to the intake. When you throttle up and open the butterfly, the other pin holes contribute more air/fuel mixture for the transition to midrange, where the main nozzle (big tube in the middle) kicks in adding more fuel with the air rushing through the venturi (carburetor opening).
I always recommend to replace the jet set if it is plugged to any extent. That is the only way to know for sure that it is clean. The inside of the tube is actually two sizes and there is no way to see if all the debris is removed from the corners where the two pieces meet.
If you purchase any new parts, be sure to look them up by your frame serial number to be sure that the parts are correct. There are some differences over the years.
Hope that helps.
Mike