a coax to hdmi converter is a good answer, so is repair the Socket and saving the best for last, buy a new TV.
Personally I would replace the female F connector. If you don't have the tools and skill to make the soldered connection, I guess by the converter.
Remember it is going to take 120VAC to operate and a means of turning it on and off.
I bought one on ebay very cheap and ran a 40' cable to the bedroom so we only needed one DVD player. Everything but the converter was 12VDC so I had to use my inverter.