Check out my kitchen in "my new/old kitchen" and decide.
We had the cabinets professionally painted , changed the backsplash and the light fixture.

I think with your budget refacing or replacing are not options. The consensus is generally to have painting done professionally to get it right so you may need to price it out. If ideally you'd do a big Reno I would probably live with it and wait. I wouldn't add a backsplash because you will have a lot more options coordinating with white and black than what you have now. I lived with a terrible kitchen for three years and then did the whole thing.

Another option to save money on paint and since you have kids would be white uppers and keep the lowers as is, and if in budget change your counter. I really like this kitchen and the backsplash is different. I think a dark counter would work on you current cabinets.

http://houzz.com/photos/6802116

I don't think you'll be happy with the look of white cabinets and your current countertops. The whites won't match and one or the other will look dirty. I would replace the counter first and see if that changes how you feel about the cabinets. I personally like them; we just bought a house with white laminate cabinets (which constantly need wiping down because white shows every fingerprint), so I envy your wood. If you do paint don't lose the grain--flat white is so easily achieved by cheaper materials. It undermines the quality of the wood you have.

Maybe if the walls were not gray the wood would look less yellow, if that is what bothers you?