OK, finally saw your pics. Go to home depo, lowes or harbor freight. You have a cheap light that was marketed for under counters an shops. The new LED ones can be daisy chained with the two prong connectors. The problem is finding a fluorescent fixture. Most things are LED now.
Look around here:
https://www.atlantalightbulbs.com/fi...cent-fixtures/
https://www.lightingsupply.com/simka...20group%20%231
https://www.lightup.com/led-single-t...d56815dd16ec1b

Lowes has an LED model like your old one that is linkable/daisy chain. I have them in my shop they are plug an go. If you look at Amazon for under cabinet lights your head will explode, there are 1,000's.

Good luck with the project!