Thanks for the serial number as it helps find the right engine model you're working with especially since 2002 was right near a big transition year of engine model change. With the help of the serial number, it shows your engine does not have the fuel pressure sensor i believed your engine to have had so please disregard my first thoughts of theory of a low fuel pressure sensor sounding the warning while reading your symptoms as that can't happen with your engine...The model design before yours had the fuel sensor i was thinking of.
I'll suggest considering with the amount of $$$ you're spending on parts that hope to work, it might be time to consider either buying or renting a marine type scan tool.
I just found someone renting the software to plug into a laptop on Ebay for $70.00 but $400.00 deposit you get back for 7 day rental...,
https://www.ebay.com/itm/284931466798?hash=item42573d262e:g:2VEAAOSwJNdi-B6m&amdata=enc:AQAHAAAAoBybZjE8lf86SYL6BhLZpBsdQ77PayFAE6YTvUaddDrnuWUyTf0FUJf7MlhK893YrtL6aTiLlBQiIfntsGZYKtYpQ2IPD8ycihQvdYjkfhuo7dDuAPjvr/qkKQowkLQHXWqfjzC4QdKeKwDQgf0/HM3REhVO9Zt+4+UJWoy20Ot9bjqMc1PFDlV+xkNSrqHyE8I6HvbJ+lK5yzjb0j6lad8=|tkp:Bk9SR4CW6NXUYA
Or depending on budget allowed, consider buying direct from Rind technologies in Chicago..., 773-736-6633
Here should be the list that would work on your engine system,
The techmate Pro is a popular one to have,
https://www.rinda.com/rindashop/shop/techmate-pro/techmate-pro-94706
Either way you'd need this adaptor to read your ECM if desired to consider this route,
https://www.rinda.com/rindashop/shop/adapter/94006
Here is your engine parts that fit your serial number for you if needed,
https://www.mercruiserparts.com/350-mag-mpi-ski-0m317000-thru-0w689999
I doubt this would be your issue you're having but your engine falls within some of it via serial number and engine model so i'll leave this bulletin here for you to at least know about it's existence...,
https://manualzz.com/doc/23430314/service-bulletin---marinemechanic.com
Also make sure you are not losing any 12 volt power anywhere under power as that could sound the warning horn too and cause a hesitation type symptom. Gently wiggle wires including the wires at the back side of the key switch.
Hope this helps some, good luck.