DVD unit says, disc is empty, with all DVDs

First, insert the DVD into the recorder.
Then, try double-click on OPEN/CLOSE button on the remote control. You have to make it very quick - two quick clicks on the same button. If done correctly, you will first see OPEN, then CLOSE in the right corner of the screen.
When done successfully, the unit will recognize the DVD again (if the burner has not failed totally.)

I tried that with RD-XS55 (it has the same DVD-RAM drive as RD-XS35.) It is not perfect solution, but it might help temporarily. In my situation, it bring my dvd drive back to "life," so I do not need to do double-click all the time.

Let's face it, your burner is dying whether you like it or not. And new replacement would cost you around $200.