The FHA route is how my wife and I bought our current house. One of the best deals going.
Some advice - get a home inspector. That's the best piece of mind I got before my 10 day window was closed. In AZ, once you put in a bid and its accepted, both parties have 10 days to walk away without consequence. It was about $300 and a good inspector will tell you everything that is wrong. Especially with a new house.
We bought a foreclosure house - a flipper from CA bought it - sat on it for 3 years then lost it. So it was never lived in. The builder did not connect the heating unit or the ducting work for the entire upstairs. There was a 3 foot hole in the garage where they put in the water heater but never patched around the tubing. They also forgot to caulk a good chunk of the outside vents and covered up some vent holes. All of that was found by my inspector.
Beyond that, if you do get a new build have someone you trust walk thru the house on your final walk thru with you to add an extra set of eyes. My friend had his electric water heater not installed just placed in its location. His dad caught it and he didn't - just an extra set of eyes. Not the end of the world, but builders can have a really relaxed schedule in deciding to come around to fix your issues. Living in your new house without hot water isn't fun for anyone.
Still to your guns and make sure the builder sticks to the contract.
My neighbor on one side had the builder forget to put in a driveway but was still strong armed by the builder to try to sign the final paperwork with the promise that "it was going to be put in.
I know another who had to visit the build site every couple of days and had to force them to put doorways, attic locations and even showers in the pre-agreed locations on the floor plans. The attic part was hilarious. The plans called for it to be in a hallway. But the contractors kept trying to put it in a kids bedroom. After fruitless discussion, a few of us went with him with a ladder, and removed the attic framing, and put it in it's proper location the day before the drywall went up.
Have fun - use your head, and definitely do as has been recommended in this thread, don't spend a dime more than you can afford. Things happen, people lose their jobs or family emergencies come up. You don't want to lose your home.
And don't use it as a piggy bank. That's your home.