In the mid 1980's, Dave Martin and his Dad, sailed a tired 25'er from the West coast to (if I remember correctly) the Northeast coast of the U.S. The boat('Direction') was so beat up, Dave had to beef up the ailing sloop to circumnavigate. After much to do, Westward Ho. I believe in '88 he met Jaja (future wife) and together they continued the circumnavigation. On the way they married and had 3 children, two of which joined as crew at the ripe old age of 1 day!
They had no where near your proposed budget, but picked up jobs doing fairly hard work in marinas and what not. That is probably not your intention, but it does point out that depending on how much one wants the 'DREAM', most things are possible.
What said, I think with a well found boat,(sounds as if you have the time to find and well prepare one, and it does not have to be lavish) with the proper safety equipment and the willingness to keep things simple, your 'plan' is doable. The older one gets, the less agreeable this sounds. I find as I get older, I object to not having ice in my beverage and I tend to shy away from the PB&J sandwiches. We all have our limits and priorities, you no doubt will sort yours out (and the wifes)as well. Learn all you can, listen to the old salts advice, maybe some sailing instruction and try to do some sailing with others. It's always good to have or make a friend with a boat. I can't tell you how many times I've read by the 'Do'ers', "Go Simple, But go NOW"! Keep the dream alive and congrates on the future little bundle.