If I'm understanding what you have written, the new DVD should hook up exactly the same way your old one did. You connect the red and white cable from the "Audio out" connectors on your new DVD to the "Audio In" connectors on the RF modulator, then connect the Svideo out on the DVD to the Svideo connector on the RF modulator. Then put your TV on channel 3 or 4 depending on what the switch on the modulator is set to, and all should work fine.
I would advise you hook your old one back up, then if it works, simply unplug the cables from you old DVD and plug them into the proper jacks on the new one. This eliminates bad cables.
If you then get sound, but no picture, the DVD may be set to progressive scan (check manual for procedure take it out of that mode),If you get nothing, then I suspect there is a problem with the new DVD player. The TV/VIDEO switch they mention, would be a switch on the TV or TV remote if you had a newer TV with RCA inputs.....You don't, so disregard that, it does not apply to your TV. You would not need the RF modulator if your TV had that switch.
Good Luck!...These things can be a pain, I hope this helps!