impulse lines cracking, or not sealing on the carb fitting are pretty common on the 031/032 saws. bogging or dying when you push on the saw is a common symptom of this(the line will seal well enough to keep pumping fuel to the carb and not leak when the mounts are "at rest" but when the mounts flex the crack opens up and you've got an air leak).
best thing you can do is to take it to a shop and have it pressure and vacuum tested like other have said. that will eliminate an air leak as an issue. parts for those old saws are mostly "no longer available" but as far as i know, seals, impulse lines and cylinder gaskets are readily available. if there isn't an air leak, then your problem is more than likely carb related. could be easy, like putting a kit in, or not so fun, like having a worn out inlet needle seat(you'd need a new carb then).