I think it's pretty realistic as long as you are realistic about the sort of lifestyle you are willing to handle. My wife and are I low-budget folks as well, don't own a lot of stuff, and live in a pretty small studio apartment, but even we have to admit that we couldn't liveaboard our current boat, though we've been toying with the idea since before we purchased her. We're also not much than you. I'll talk about our biggest reasons for choosing not to move onto
Essorant, and maybe when I'll have more time I'll talk about specific boats that I would consider moving onto.
Make a realistic assessment about how much space you'll need. Sure, you're crazy about each other now, but once in a while you'll want some privacy. You can't get that on a 27' boat. Heck, you can barely walk past one another.
Another big deal for us is hygienic stuff that we take for granted on land.
Essorant has a tiny head with no sink and certainly no shower. Imagine winters in the PNW, hiking up the dock to the marina's shower facilities.
Similarly, try to figure out where you're going to keep your stuff. You can only minimalize your lifestyle so much. You need two sets of clothes each -- one for summer, one for winter. In principle I can see us storing all the clothes we need in one of the quarter berths -- if we had some place to put the sails that are in there now -- but even then, it would be such a PITA to get at them. Finally imagine how you'd have to rearrange everything in your home every time you wanted to go sailing. It would probably keep you on shore.
Also, what happens when you have to haul the boat out?
So yeah, space to live, space for stuff, and showering in the winter.
Now, at the boat show last weekend I saw a big improvement in the jump to, say, 35'. Builtin showers and walk-in quarter berths with double beds. Convert a quarter berth and the v-berth into storage, and you've got yourself a liveaboard. Maybe you could get that from a 32' boat as well.
One things we're currently thinking of doing is moving into an even smaller (i.e. cheaper) apartment mostly as a place to keep our stuff and have a shower, and maybe spend half our time on the boat. In the summer we could spend most of your nights on the boat and still know that we've got a dry home to go back to. In the winter we would live in the apartment.
The biggest different between our scenarios was that I didn't have any sailing experience when I bought the boat, so my plan was: learn now, liveaboard later. You're maybe already past that stage. Still, if you have a sailing bug that you want to satisfy -- and your girlfriend has caught it -- consider spending that $5000 on a boat that you don't plan to live on, until you can afford something that you could see you cramming your life into comfortably. Take the $7000 and put it in the bank and don't touch it. That's more or less what we did just over a year ago, and I'm happy with the decision and now looking to upgrade to a liveaboard cruiser, maybe two years from now.
Regarding being too young -- use it while you've got it

Just be rational about it and you'll thank yourself later.