Sorry I can't give you an exact date your engine was built but jugging by the serial number, it looks to be in the 2001 era. Typically when it comes to finding the correct parts for your engine model, Merc doesn't really go by year but rather engine serial number breaks/ranges such as you'll find in the following parts catalog for your engine serial number provided above...
http://www.mercruiserparts.com/selectSerailRange.asp?doc_nbr=5.7L+EFI++GM+350+V-8++1999-2001
Did you find the number on top of the engine or down by the starter like in the photo above? And do the numbers match as sometimes they'll change the engine block and reuse all the bolt on accessories to make the engine function.
If you really want to know the exact date of the engine you can either call a Mercruiser dealer with the serial number or call/send an email to Mercruiser directly in the following link,
https://www.mercurymarine.com/en/us/contact/mercruiser-customer-assistance ...Just hope the engine wasn't reported stolen.
:gossip::mod::bolt:
Unfortunately that sensor under the distributor can be sensitive to many things and typically there really isn't an indentation of a failure about to happen as most times, It just goes....Some have reported that the engine may first run rough though. Also there's a chance that the sensor was just lacking quality and went out on it's own as yours wouldn't be the first to read on the boating forums that this sensor went out in a short time frame and they installed a new OEM one and has been fine since. That mentioned, it's advisable to carry an extra one onboard along with the tools to change it if ever needed...As you found out/mentioned above, one of the quickest ways to know if it's working is to use your test light connected to 12v power and touch the negative side of the coil and see if it's blinking or not at engine crank speed. Here's an old example for you but keeping in mind that it's just out of your serial number range.
Sensor Failures in MEFI-3 Mercury Distributors Used on V6 and V8 305/350 cid Engines
It's possible the coil not installed properly could have played a role in shorting out the sensor had it touched something or got/gets excessively hot.
Be sure to check that all power and ground cables/wires going to, from and on the engine are clean and tight.
Ensure the distributor housing assembly is clean and seated all the way down to the intake manifold and tight. Ensure the distributor shaft isn't excessively loose, bent, wobbly, etc..
Install a new rotor and turn the engine over checking to ensure the rotor wheel isn't hitting the sensor as it passes through.
Mercruiser recommends putting a small dab of blue Loctite on the shafts keyway and or locating key of the rotor wheel to help keep it from lifting at high RPM. Page 4B-6,
http://www.boatfix.com/merc/Servmanl/16/16B4R2.PDF
The latest sensor and rotor OEM part numbers are 13524T1 & 87-892150Q02 or item numbers 5 & 6,
http://www.mercruiserparts.com/Show...=80&bdesc=Distributor+And+Ignition+Components
Be sure to tightly connect the black sensor ground wire to the rear of the engine ground stud where the battery cable normally connects to... Doing this should give you the best results plus it will give you the chance to inspect the battery ground cable condition.
Hope this helps, good luck.