Honda 4514 lawn tractor problems?

Question:Symptoms: first, stalls when eng. cold. Restarts fine. Occas. stalls even after warm. This year, now the eng. runs rough, has low power. Uses fuel at an high rate. Eng. smooths out somewhat when engaging mower blades, but has little power.
Things we've done to try to remedy: checked air filter-good. Checked fuel lines/flow-good. Replaced coil with new-good. Replaced carb with new-no change. Replaced spark plugs- old plugs were fouled with dry black carbon-no wetness. Fuel pump seems fine-when turning ignition to 'On', can hear clicking as pump fills carb fuel bowl. Good fuel flow thru fuel line (took off line, and discharged fuel into pan).
Once started the engine runs ok for about a minute, then starts to stumble (rpm drop). Governor starts hunting to adjust the rpms up, down, up, down.engine stumbling all the while. If we clamp off the fuel line partially with a hemostat, after about a min., it smooths out. ??! Also eng blows air out carb. Any ideas? We're stumped.

Answers:
Timing may be off or valve problem. Check compression. Remove the flywheel cover and check if keyway is still aligned and key is not damaged.