You are exactly right - it starts with a hum, then eventually will short cycle the burner so that the boiler never gets to its maximum temperature.
So, if I understand you correctly, the Beckett is not a direct replacement for the Honeywell.
Long time reader, first time poster!
We have an old American Standard natural gas boiler from when the house was built in 1967. We've owned the house almost twenty years and in the last eight or ten years I have replaced the Honeywell Aquastat (L8148E 1265) no less than four times. It seems...