Here is the situation you are dealt with. Most likely the GPU on your Xbox360 is messed up, which is the cause for the reason you have no video but audio when you load your Xbox360. Gamestop will accept your Xbox360, however, since it is not in full-functioning condition, they will charge you a refurb fee.
I believe the normal price of a traded-in 360 is $50. Gamestop charges you a $30 refurb fee for any broken console. So you would only get $20. If you're lucky, there is a $25 bonus trade-in credit if you trade in your Xbox360. The most you would get is up to $50. As for attempting to trade in your broken Xbox360 toward the new Slim bonus ($100), the gamestop representative I just talked to said that I would get little to nothing since it was broken, but you should try to contact your local gamestop and figure out.
If you somehow get your xbox360 working, you will get $75 trade-in credit if you don't trade in the xbox360 toward a slim or $60 cash. If you trade it toward the slim, you already know the trade-in offer details and how much you'll get for that.
As of an hour ago, I just bought a new arcade console since I thought I would try to wait out the slim and see how good it was or not. The console was $150 that included $50 gift card. I am planning to sell my Xbox360 arcade console to Gamestop for at least $60 cash. Therefore, I am really only paying ~$40 for the Arcade console disregarding the $50 walmart gift card. You might want to take a look into this route since, say you do get your xbox360 working, you would still have to pitch out like $200 to buy the slim since you only get a $100 bonus if you trade it into gamestop.
Your two choices are either the arcade or the slim:
$90 (includes $50 gift card) or $200 slim
Of course, you need a working xbox360 to fully benefit from this. For a temporary fix that might make your video work, the towel trick may work. I am not recommending it though since it's illegal cough cough.