I have a similar problem in my yard. Manually pulled all of the dandelions down to the roots 2-3 times in the past month. Sprayed some more weeds in the front but couldn't fertilize because a big portion of the front lawn needed to be reseeded. The back was even worse - so many dandelions that I couldn't pull them. Sprayed with Weed 'B Gone 3 times now and they keep coming back. Wanted to fertilize, but again, I have a large portion of lawn that is being seeded (old owners had a pool).
Didn't even think about a dandelion puller tool. Maybe that's something I can convince the girlfriend to help with. Now I just have to find one.