Small Engines (Lawn Mowers, etc.): sears riding lawnmower problems, float bowl, fuel demand

Question
sears LT/10 model 502 250840 date 7108 The problem is:it will only run with the choke on.I have thoroughly cleaned the carb using a piece of tag wire and carb cleaner.the float bowl screw has to be messed with constantly to keep it going. it will run then the lawnmower will start to die and I have to mess with that screw and the cycle continues. If I try to hit the gas it will try to stall, I then have to hurry to hit the choke to keep it going. and I have added FRESH gas  

Answer
I hate to tell you this, but the carb is still dirty.  It's not uncommon for missed debris in the system to contaminate the small orifices in the carb.  It's also possible that you missed one passage.  The reason it is surging is when the governor spring opens the throttle, the fuel demand increases and when it can't be met (because of carb problem), the engine dies back down.  When it dies down, the low speed circuit comes into effect and the engine has enough fuel to idle but the governor spring once again opens the throttle and the cycle goes on and on.  It probably idles just fine though.  Because you have cleaned the carb and it has this problem again, I recommend you also remove and clean the fuel tank and check your gas can for debris as well.  Hope this helps.