Don't go spending money on a new carb just yet! I'd be willing to bet that you have a governed carb on that saw. Both of my 181SEs and my 288XP have them. There's a big ~1/4" dia. brass screw on the side of the carb right near the choke linkage. Unscrew it - be careful though because they're threadlocked into place and it takes a bit of force to break the threadlocker, and it's easy for the screwdriver head to slip and strip the brass screw. Best to use a fresh flathead screwdriver that isn't rounded over on the end and that fits the groove nice and tight.
Once you get that screw out, you'll be able to see the governor circuit inside the carb and the ball-bearing-and-spring valve mechanism in the screw. The way these work is that after a certain RPM, the spring will compress and open the valve, dumping fuel into the carb throat and effectively flooding the saw a bit. If this valve were to stick open (or spring get too weak) it could definitely flood the saw at lower RPM or after shutoff, I suppose.
If you know what you're doing with tuning by ear, and it sounds like you do, cut a small 1/4" disc from an aluminum can or thin sheet rubber, drop it into the governor hole, and insert the brass screw on top of it. This will block the governor circuit and allow you to tune it like a regular carburetor.
To retune the carb from zero, turn the screws in to where they seat, then out to 1 turn. Start the saw and make sure it'll hold an idle by adjusting the idle screw in. Then tune the H jet to where you get a slight four-stroke sound. Then move on to the L screw - move it just out to where you get a good throttle response and acceleration from idle to wide-open. Make sure the High tuning is still good by opening it to WOT again (sometimes tuning the L jet affects the high end a bit too); then turn the idle adjust screw to where it'll hold a nice even idle without stalling or moving the chain.