My Atari 2600 doesn't show a picture

[masschamber]

Try an RCA to coaxial adaptor, they are much more reliable than the rf switch boxes,
http://www.atariage.com/store/index.php?main_page=product_info&cPath=21_28&products_id=42
though they can be found dirt cheap on ebay, and you can make your own in about 5 minutes with a solder iron
you can also do an rca mod
http://benheck.com/book/support/Atari2600VidMod.htm
I've done this it is easy

for NiKKoM suggestion
http://www.atarimuseum.com/fb2hacks/
it is very easy to do, I've done this as well
and you can use the parts from your original console for the cartridge port if it really is shot
using the flashback 2 modified also works well for portablizing

personally I've still got a heavy-sixer (the original 1977 units made in Sunnyvale, CA) hooked up and running, it still works perfectly. Out of the 30 or so 2600's I've had over the years only 1 Jr. model didn't work (it clearly had been exposed to extreme temperatures), so I'd be surprised if it is actually broken
I'm almost positive it is a bad switch box judging by the symptoms you described