The rhythmic rpm shift can be caused by varnished fuel components in the carburetor, which leads to lean mixture. As the engine begins to stall, the governor spring pulls the throttle back open, increasing the rpm's, and the cycle repeats. You might be able to cure this by adding some carb cleaner to the gas tank. Or, you might need to get the carburetor overhauled.

thanks for the tip. i was sort of afraid it was going to be the carb. i added stabil to the fresh fuel just now and according to it - it has some sort of cleaner in it. is this what you are talking about to try first, or is there a different carb solution to be added to the fuel? beacuse the stabil thing didnt really work - it almost got worse. i saw at home depot a spray can of something called "carburator cleaner" but i dont think that is to be added to the fuel - more like to spray somewhere, or possibly use during carb rebuild?

second question, if I were to rebuild the carburetor - what would you suggest? I've never done it, how hard is it to figure out by following online instructions? i just looked up parts - rebuild kit is $15-20 and time it takes to learn and do it. the new carb is $150-$160 but could just plug and play. if i were to take on rebuilding it myself - would i need any specialized tools? if i were to take off the carb and take it to someone to rebuild it - how much would that typically cost? thanks again - trying to weigh options as the branches are piling up...

Common problem when engine sets alot and occassionally used. Carb gummed up probably as suggested as above. I would first try to use fuel additives to clean the passages. If that don't work I would then remove the bowl on the carb. and spary (carb. cleaner using the red nozzle provided and safety goggles) the needle and seat any small passages you see and up throught the middle. Last resort IMO would be to dissassemble the carb. clean all the find passages.

Removing the fuel bowl on the carb. not that big deal providing you can get it back on without pinching the gasket or o-ring creating fuel leak or over tighten the bowl mount bolt smashing the gasket or the bowl creating leak. Again I would run additive through it first (gum-out if you cheap, more costly are seafoam or lucas which work better IMO. Would be good idea to change the fuel filter and use additive regularly especially if you are running corn gas!

As for the backfiring, are you bringing it down to idle and letting it idle for awhile before shutting it off?
This might help, it only needs to idle down for around 30 seconds.
As everyone else said, for the surging while running, most likely some varnish in the carb. Try cleaning it first before attempting rebuild.

i realize it has been almost a year since I posted this and I never did say thank you for the great advice given. So thank you. I bought a can of "Mechanic in a Bottle" from Home Depot ($5 i think) and used that as additive. after about 10 minutes of running on that stuff, it cleared up and was running steady RPM. been using stabil since then in the fuel. after sitting for 8-9 months and starting up - it was doing it again. but this time after about 10 min of running with just stabil in the fresh fuel, the revving went away. Thus thanks to everyone. though it still backfires every time I shut off the engine. There is no option to rev-down the engine, just a switch start-stop, electric engine starter button and that is it. no throttle control.

First-
"briggs and stratton 18hp twin cylinder vanguard, model 422437 type 1899 E1."
A 422437 is an opposed twin. Vanguards are OHV engines, not flat heads!

Does the carburetor have a fuel shut off solenoid?
They are designed to stop the fuel flow through the Main Jet to prevent afterfire (backfire).
If it does, is it intact?