You haven't given us enough information to give you reallly specific help, but we might be able to figure out a few areas you should think about.
First, you should be able to get good, if small, shots from the Sony. If the 5x7s from WalMart or Target look bad, then there's a good chance that your technique is poor, and before you do anything, you should get a good solid how-to book from the local bookstore or Amazon. If that's the problem, ask here and you can probably get some good book recommendations. For example, when shooting kids, even when they're moving, the camera can't be jerked, any more than you can jerk the trigger on a rifle. You'll get blurred shots. If your problems are more in the area of lighting, then look at a lighting book -- you may simply be standing too far back (using too much zoom) for the reach of your flash, which is quite limited.
If you really do need a good step-up camera, the Nikon D50 would be fine, unless...
The printer you are getting could outrun the resolution of the D50 at the sizes you mention in your post. If you shoot your D50, the crop a photo and print it 24 inches wide, you're going to be disappointed, because the pictures are going to be blurry.
Also, I would urge you to do further research on your printer, because I think you are about to make a mistake. If your grandson is about to be born, then the prints you make on the Designjet may not last until he's an adult. The Designjet uses dye-based ink, which even when "fade-resistent," WILL fade. A better choice for you might be the Epson 2400, which will give you a wider print that you can probably use with an entry level camera, will cost half as much as the Designjet, and will have pigment-based inks that will fade much less (we all hope.) I suspect that most people on this forum use Epsons, although I may be wrong about that.
And finally, you suggest that budget isn't much of a problem. I don't know that that means, but if you are reasonably affluent, you might consider buying a higher-end camera than the D50, such as the Nikon D200 which is coming out this December -- next week, I think -- and which will give you something close to the best digital shots available. You could buy one, I think, for the cost of the D50 plus the savings from not buying the Designjet and going with the Epson. And THEN, use any extra money to go out to the Santa Fe Photo Workshops, which you can Google, and take their really excellent course for new camera users, which is designed for people exactly like you. It's a week-long course, and when you finish it, you'll know how to take good photos, even if you'll need a little additional experience before they become really good. You'll also get exposed to the Santa Fe workshop culture, which can provide courses with well-known photographers who will take you as far as you want to go.
JC