The problem may be somewhere with the backlight wiring. It would be best to open up the Gameboy & double check all connections now. Check to make sure that the wires are soldered on good, & not loose or barely touching in any way. Also, make sure that connections aren't touching each other. Examine any new solder points & make sure that they aren't touching anything else that they could short circuit as well. This includes when you close the case. As you are closing it & putting it back together, make sure that any solder blobs aren't so big that they'll touch something they're not supposed to when you close the 2 halves together.
Another thing to do is to clean the battery contacts. You can actually pop them out of the case to soak them in battery cleaner or something, though you don't absolutely have too. Make sure you scrape the surfaces that touch the batteries clean, even if you do or don't soak them in cleaner as well. Feel free to test it before you even do anything. Does the system short out like that if you touch, or even spin the batteries in place, while it's on??? Then you definitely need to clean the contacts. This includes the tops of the springs as well.
BTW, don't worry about slow responses. On forums like this, it usually takes much time to get a response, sometimes even a day or 2, because not everybody is online all the time, especially somebody who would know specifically about your concern. A display of quick frustration will definitely attract the trolls. Do not invite them your way. Trolls are everywhere. ^_^
