Well, saturated it ain't. That doesn't appear to be the problem. To me, the problem doesn't appear to be saturation, so much as color shift. Your printer evidently has a setup with a strong bias to yellow. Both of those samples look like a yellow filter was applied to them. And I'm not sure that RGB vs. CMYK would fix that.
I doubt that your files are in CMYK, though, since, in a quick test, attempting to export a CMYK file as a .png gave me a file that ended up as "Indexed Color" rather than a CMYK .png. I suspect that .png may not support CMYK. Or at any rate, not from Photoshop CS 6. (Given that it's a *network* file format, it probably isn't designed for printing.)
Both .tif and .jpg support both color modes, I'm not familiar with Wintel color modes, like .bmp(?) if anyone still actually uses that, so I can't answer for those, but I am sure that at least some of them do. Since one huge advantage to .png is the ability to produce images with a transparent background, if you don't need a transparent background you might be able to consider trying something like .jpg, although it's lossy if you have to resize, and I'm not sure it would address the problem either.
Going from screen to print is always going to come with the possibility of a nasty surprise at some point down the track. What you see on the screen is pure RGB, and it has a far wider gamut than any kind of printing ever will. There are colors that simply cannot be duplicated with ink. And trying to adjust the file in hopes of compensating for the printing is always going to be a stab in the dark.
Converting a file into CMYK and converting it back into RGB can get rid of some colors in an image which are outside gamut for CMYK, because data which is lost with the first conversion does not return with the 2nd. But I'm not convinced that is going to fix the problem of too much yellow.
I submitted my files to MPC in RGB .jpg and was fortunate, since MPC did an excellent job with them. But then, my files didn't have large areas of flat color, as yours do. Yours have much less room for error.
Q: are you scanning the black and white drawing and colorizing it in MS Paint, or are you colorizing in analog and scanning in the colorized version? Does your scanning software give you any color correction latitude, or other features? The more correction you can manage to do *in the scanning* the less you will need to twiddle with in the application that you use to edit the image. Editing loses data. Scanning collects it. If you collect more in the first place, you have more to work with.
Is there a color balance feature anywhere in either your scanning software or your image editor? The opposite of yellow, in RGB is blue. Nudging the image toward blue might concievably help, but it might also just make the mess worse. How much time and expense are you prepared to devote to this? You might do better just finding a different printer, or asking the printer to correct the excessive yellow.