OK these things are stone simple, typically if it kicks out its due to either a damaged or dirty CAD cell (the little light sensor thingy) or the ignition isnt actually firing consistently or the gap is too big. Running a worn out nozzle will sometimes not start, and it will soot or burn extra fuel, but unless its totally junk or clogged it should still run with a worn nozzle.
So first the troubleshooting. Crack it open, replace the nozzle. This is just for a PM, not really needed in most instances but nozzles are cheap and its good to have a new one before fixing everything else. Note that there are different styles of nozzles and its furnace/combustion chamber specific. RTFM and replace, no guarantees the one you took out is correct.
While its apart, look at the two electrodes that ignite the the fuel. They should have pointed tips (like a pencil) and be gapped properly. If they are worn out, replace them. If you need to "get by" for a weekend, clean them good with brake cleaner and resharpen the tips with a grinder. But replace them ASAP. You don't need need the Beckett gauge. New electrodes come with the gauge but you can set it by eyeball and using a caliper or a few drill bits if you are handy. Its just gotta be close, doesnt have to be micrometer-precise.
To diagnose the CAD cell, jump across the wires and fire up the burner. If it runs happy as can be and doesn't shut off until the thermostat commands it, you have a loose wire, dirty, or damaged CAD cell. They rarely die, but it can happen. You can also have a CAD cell giving a higher resistance if the flame is weak, you can check them with a multimeter.
Lastly, make sure the connections between the electrodes and the igniter are nice and clean. Occasionally my igniter would fail to start, I would go clean the contact springs up with some brake cleaner and scotchbrite and put it back together and it would run for another year or 2.
Then when you are all said and done, whip out your manometer and set the draft.
https://falconrme.com/wp-content/uploads/2019/06/Beckett-Diesel-Burner-Manual-English.pdf
https://www.acservicetech.com/post/troubleshooting-a-cad-cell-flame-detector-for-an-oil-burner