The Tx and Rx LEDs are not connected to the 328P but to the serial-to-usb converter (16U2 on the official Uno, CH340 on most clones). So I suspect that that converter is malfunctioning.
If your board does not use the CH340, you can run the loopback test. If that fails you will know that burning the bootloader to the 328P will be a futile exercise as it will not solve the problem.