Fluorescent lights sometimes don't turn on
Do not use plug-n-play LEDs - they depend on a functioning ballast and your ballast is suspect.  

Because then, you'd be at a crossroads.  Return the plug-n-play and remove the ballast and get a direct-wire/ballast-bypass LED... Or replace the ballast and stay with PnP and support that ballast forever, even though LEDs don't actually need it.  Obviously the second one is pretty lame.  

I honestly recommend a third choice, replace the ballast and keep that nice new tube you just got, that is probably 90 CRI and better light than you'll get anytime soon from LED.  Modern ballasts fix all the complaints about fluorescents.