I don't think that that is a healthy sign. TX can light up if the loaded code is constantly printing. RX should not light up unless you have something sending data to the Arduino; if nothing is sending, in my opinion that would mean that you have (partially) damaged the serial-to-usb converter.
Which component?
Please explain what that means. Does the sketch that is currently loaded send data to the laptop and you can see it in e.g. Serial Monitor? Or does it mean that your operating system still recognise it? Or ... ?