Yes you will have to take the carby off and clean it thoroughly.
I am reminded of the many issues plaguing the old Zenith carbys on Southern Cross (and some Victa) mowers. Generally the problem is the initial valve seat, in which case either: 1) no fuel will be getting into the bowl at all if it is blocked or the valve spindle is being held in an upright ("closed") position due to dirt or wear, or 2) too much fuel will be getting in and flooding the carby. Again this can be caused by dirt/blockage or just plain wear and tear. Sometimes a small amount of hydrocarbon sprayed on the valve will help free it up again. Try a simple gravity test. When you inspect the valve, try tipping it up and down and watch to see if the valve spindle is sliding up and down (or in and out) under the influence of gravity. Also, there is sometimes a very small sealing washer in there that may be perished or missing altogether. If this has broken down it may be what is blocking the valve.
Another problem is with the floats. I could not work out why one of my carbys kept flooding. One day I inspected the float and noticed a tiny pinhole. Placing it in a cup of water I pushed it down and sure enough it started to release a string of tiny air bubbles. In my case the float was made of brass so it was a simple matter of soldering up the leak and sanding the patch down with some fine emery paper until it was smooth again.
Take the main jet out and make sure it is clean. Blow air through all of the jets and holes in the carby. Use a very fine length of copper wire or a pin or needle to clear any blocked holes, but be fairly gentle about it.
Occasionally if a carby gets badly worn it will start leaking fuel out of some of the hinge and screw holes. In this case you may need to seal the bottoms of these holes on the outside of the carby with something like Pro-weld or JB Weld. But make sure you do not interfere with the movement of the hinges and you can still take the screws out.
Good luck. These are just some of the issues I have had with old carbys and thought they may help in your case. It is really not that unusual for an old carby to leak a bit of fuel when you first try to start the motor. Generally this is not a big issue unless the fuel continues to leak once the motor has started and the choke has been turned off.
