I think in your case a complete bike is the way to go. It just seems silly to me, especially when you have a tight budget, to throw away $50 - $150 (your estimate) on a build-up. It's also an unfortunate fact in the bicycling world that component groups and individual parts are almost always way more expensive when bought without the bike. So, you're already losing money by buying the components outright, and then to pay for the build? Waste. Start with a solid pre-built bike and as you come into more money over time, upgrade from there. This way, you'll be able to get going right away with a bike that willl serve you very well for the time being, even if it's not your perfect "dream bike". Also, in the meantime, while you're upgrading, you can also learn how to turn a wrench while installing your new parts.

Another bonus (and you might not be concerned with this type of thing): you'll probably end up with extra parts, like an extra wheelset for example, that could be used during winter months (it's great to be able to swap wheelsets with different tires, depending on conditions).

I just know from experience: I built up a Cross Check for myself and it was a very expensive endeavor. The frame was even free (a car hit my old bike, driver's insurance bought my new frame).