Your system still works fine I'm sure. It's just that the pin connector inside that the game touches when you put it in is old and corroded. This is a problem with a lot of old systems, and probably more so for the NES than any other system, the way the game has to touch the pins just right at an angle.
My suggestion would be to buy a brand new 72-pin connector off of eBay. You can find them for under $10, and will be much cheaper than going to buy a whole new system. They are really easy to install. Some even come with instructions.
Just search for Nintendo 72-pin connector and you should have no problem finding one.
Once you install the new pin connector, the games should fire right up, unless they're dirty. Just get a q-tip and get it damp and start scrubbing the pin connectors on the bottom of the cartridge. You should see plenty of dirt coming off. That means it's working.

You may even want to try cleaning the carts before buying the new pin connector. The one in your system may not be 100% responsible for the blinking screen (although it won't hurt to change it in the future).
Good luck. I hope you get it going!