ReMorton,
Plastic cars, boats and motorhomes can be a bugger when it comes to grounding circuits, and especially over time thanks to the physics of electrical flows, the process of electrolysis can oxidize and/or create corrosion on, around, under the connectors and posts, and within the strands wires themselves, creating increased resistance or eventually a lack of ground entirely.
So if it was me, I would start at the battery (-) post and follow the ground path to where the alternator is grounded, likely the engine block, and inspect each connection and cable for oxidation. remove connectors sandpaper or wire brush the parts to remove any chance of oxidation or corrosion causing resistance or breaking the circuit. As I connect them together, I always try to use a AntiOxidant when ever possible on connections (especially aluminium connectors), even if it's just wheel bearing grease, even better fancy stuff like this:
1oz. Antioxidant by G B Electrical Inc - OX-100B - More Wiring connectors and terminals at doitbest.com
Then next follow the path from the output (+) of the alternator back through all the crap to the battery (+) post. This will assure that all connections are not oxidized to the point of creating enough resistance to not recharge the battery even though the alternator is working.
Are you using a traditional Ford wheel-well mounted regulator? Or using a one-wire regulator like the Transpo F7078 which mounts to the rear of the alternator? I use the F7078 and have not ever had a problem in 13 years, although I rebuilt my Motorcraft alternator this summer due to blown diodes. If your using the one-wire regulator double check how it's wired in. If you are using the traditional Ford regulator, do the same wire path investigation I recommended above but include the regulator too.
Finally how old is your battery and is it in good shape? Unless you consistently use a battery float charger during the months that a Western Washington Cobra sits in the garage, from personal experience I find that the battery ages quickly, much quicker than normal. Boats and motorhomes are the same way. My Corvette is horrible without a float charger the battery goes dead in a couple months of sitting. Older batteries in cars that sit for long periods un-driven and not on a float charger, can appear to be fully charged showing 13+ DC Volts at rest, but when it's time to put out the AMPs needed to spin a cold high-compression big block they choke and puke.
Sounds like the alternator has been tested for output on a bench, dumb question but have you done a stress test on your battery to see how many AMPs it will produce and how long it will hold the AMPs it does produce? This could be another issue creating a 'charging' issue where the alternator and regulator are fine and the battery just does not take the deep charge anymore, or the battery is fooling the regulator with the correct voltage range making it think is is fully charged, yet no cranking power when the key is turned.
Nice rainy weekend ahead for working in the garage on the Cobra, that's my plans.
Cheers!
BC Kevin