If you are happy with the vanity itself (just not the wear and tear), you could save that with some TLC and fairly inexpensive treatment including paint.

I just did my kitchen with chalk/clay paints. Some manufacturers are Annie Sloan, The American Paint Company and Ce Ce Caldwell (which is what I used). I purchased paintable wallpaper that looks like beadboard, put it on the facing of my kitchen cabinet doors and drawers, then painted with the chalk paint. These rolls of paintable wallpaper are at Lowe's and Home Depot, I think they are $19.99 a roll. You could also use this around the room like wainscoting, which is an idea I got from a Houzz article a while back. You then attach molding around the top on the wall, and you can't tell it's not actual beadboard paneling.

Link to the pictures of the beadboard wallpaper painted on my cabinets here (click on words under picture)
http://www.houzz.com/ideabooks/12451092/thumbs/Kitchen-upgrade

Depending on how hard your kids are on things, I'd probably paint the vanity cabinet white as it is but use the chalk/clay paint. It already has a distressed look around the doors and base, so that's great. This paint is really easy to work with and doesn't require you to sand the item being painted.

I'd even keep the blue around the vanity top and on the mirror and use a somewhat subdued nautical theme.

The biggest expense is going to be if you want to replace the flooring. Tile and grout aren't that expensive, but if you are reusing everything you already have, if you aren't handy maybe you can pay someone to do the tile work for you? I'd just go with white. I'd probably also try to match the blue in the mirror/vanity and paint the outside molding of the shelf unit over the toilet with as close a match as possible. You could just buy one of the $3 sample sizes for that.

With the above you are talking about maybe $100-$200 for paint, maybe $40 for 2 rolls of the paintable beadboard wallpaper. Not sure what the tiles and grout would run. You'd have to search your local stores for the chalk/clay paint. They aren't often at the big box stores, but you can order it online from the same place I got mine (link below) or search for one near you.

http://fabulousfinishes.wordpress.com/

Good luck. Looks like a fun project!

Can you DIY? A vintage dresser cut out to fit two new sinks/faucet would look just right with your shabby chic desires. A new floor (even linoleum) would freshen the space up. Crisp white towels, a white waffle shower curtain will make the room feel "spa like shabby chic".