Hello! I am having a crisis!

So, the other day for three bucks, I picked up an old bathroom vanity fixture form the 60s at a local thrift store. I incorrectly assumed that it was preheat, because of age, only to discover that it was trigger start. Well, the ballast in that ole' thing wouldn't start the tube because of that old fouled capacitor, so, confronted with some unusable PCBs, I did the only reasonable thing and went shopping at our local farm supply store, and picked up a brand new 14-20 watt magnetic ballast, made for preheat. Well, I sure got the right part because it literally just bolted into place -- there were some little cutouts in the metal from the factory and I guess that they have been making the same exact product for all of 50 years, and I was just so pleased. What  I neglected to purchase was a starter socket, so I just soldered the leads directly to the starter. This melted the plastic, but didn't seem to do any harm otherwise. Well, I fired it up, and I let it run for about two hours, and all was well. Then I did it again and, confident with my new contraption, went outside. I came back in only to discover that the starter had re-engaged whilst the lamp was still lit, and the extra current generated by the heating of those filaments, besides any adverse effects to the tube, had heated the ballast so that it scorched my wooden table! A one off deal I figured but no! After a few more tries, the tube didn't like to start, just flashing with that starter and superheating the ballast. When it did start, the starter would soon fire and the it would try to re-start mid-operation! The only reason I could see for this happening would be that perhaps I damaged the thermal properties of the starter whilst soldering the leads directly to it, so I replaced the starter, and got a new socket for it too. By this point, I am $6 in for a new tube, $8 for a starter socket (where do you get affordable ones?!?!), $13 for a new ballast, and about $9 for two starters. And the problem persists. The lamp is very hesitant to start, merely flashing for a period, ie failing to start, and if it does, it usually does not last, falling back into the fruitless starting cycle. I am confident that I have the correct ballast, tube (F15T12), and starter (FS-2, rated for 14-20 watts). All components are new.

Please help!

Sage

I've found that some modern tubes don't work all that well on preheat.

Try removing the starter as soon as it starts & see if it'll stay lit...if it does, leave it on all day to give the tube a good burn-in period. It should (hopefully) be easier to start after that.

I agree, sometimes with modern tubes, they'll need a few hours to burn in before they play nice with preheat. Sometimes I have to run it on another trigger start fixture first or manually preheat the first start up, at least until the lamp has ran for a few hours first.

Starter sockets are pretty hard to find still in stock in stores, you might have to order them online. These are probably the most affordable starter sockets online nowadays.

I doubt the original ballast contains any PCB oil, since trigger start ballasts rarely contain a capacitor. They are usually a simple core and coil design.

Something definitely sounds off. If you have a Kill-A-Watt meter, see what amperage and voltage it is drawing. If you don't have that, try a different tube. It is possible that the ballast isn't good. What brand and model is it?

You may go for an electronic starter.
Or if the place allows, a manual start (if there is a convenient way ro mount the starting switch).

You could always test the components in another fixture if you have one that's suitable.
There is a chance that the starter may have gotten stuck during testing. This results in only the filaments glowing and the ballast getting warm. Leaving the fixture running in this state will either fry the ballast or cause one of the filaments to break, depending on what happens first.

I recommend getting the ballast changed out to a brand new preheat ballast.

I recommend using a 13w PL-S choke ballast since it drives a F15T12 fluorescent tube close to spec as some advance 14w-20w preheat choke ballasts will overdrive F15T8 and F15T12 fluorescent tubes quite considerably.

Here is a 13w PL-S choke ballast that might work for you:

https://www.ebay.com/itm/285086867512