Your fixture needs a new ballast.

And a new starter too, while you're at it.

Thanks for your assistance so far. Will it be easier/cheaper to replace ballast and starter or just replace the entire fixture?

You could check, but ill bet the entire fixture and you will get new sockets also.

Installing bipin fluorescent tubes with one pin outside the socket- easily done unawares- will soon cause the problems described, but not that quickly. Ballasts do unusual things. They provide a voltage surge, perhaps 600 volts for some type ballasts, to get current flow started in the lamp and then they limit the current flow to the correct amount. Once started, the nature of the lamps is to permit virtually unlimited current to flow through them. That would result in meltdown or explosion. Perhaps yours approached meltdown.