OT - Flurescent light problem

Hey Ralph.  Just a couple of ideas.  First, highly encourage switching to LEDs.  Much less finnicky.  If the circuit is on a dimmer switch, make sure the dimmer can operate fluorescent bulbs.  Make sure the bulbs are dimmable too.  If you switch to LEDs, make the same two inspections:  check the switch and check the bulbs.

I had a situation in my kitchen several years ago where two fluorescent bulbs kept burning out over and over.   I even hired an electrician to troubleshoot - no luck.  I eventually replaced the dimmer switch and everything has been good ever since.

@Ralph Lipeles,
Does that single ballast drive both bulbs?

Added later 04 min 36 s:

@Joe Fleming,
I agree. Low voltage, much higher light output using a fraction of the energy. I've replaced about everything with them except my shop lights. I have them, but have been too lazy busy to install them.

Yes, search for Circular LED light replacement or similar. Might be easier to change whole fixture but that depends, of course.
Other option is to get a LED "rope light" and coil it inside the fixture but it likely requires it's own low-voltage power supply box which would be a problem.
Probably lots of options but challenge is finding the right one, especially if dimming is required.

@Ralph Lipeles,
Based on your description, it seems the new ballast is bad, or has gone bad. Since it was never able to fire both tubes at once, and then started "sparking" after continued use, I would suspect something in the circuit such as bad connections. Since it seems to follow your swapping connections, maybe the ground that failed over the years?

Be careful working on it. Ballasts hold a charge of tens of thousands of volts that will definitely knock you off a ladder or chair. Touch the ballast wires to ground to discharge them ensuring you are not in that path to ground. And then there is the issue of switches being wired incorrectly where the neutral is switched instead of the hot, meaning the hot wire to the fixture is hot even when the switch is off.

Ralph Lipeles wrote:

Do they make LEDs in a round configuration so that I can redo the ceiling fixture?

Not a problem. If you do a search on Amazon for the words "round kitchen led light" you will likely find more than a few that would fit the bill. I replaced all of the incandescent and fluorescent bulbs (with the exception of my oven and clothes dryer) with LED fixtures and bulbs and the difference is amazing. Using less than 1/4 of the power they give at least as much light although I'd suggest that, if you intend to make such a change, to read up on color temperature -- LEDs generally come in three different ranges and some people find that having them mixed can be annoying from an aesthetic viewpoint.