Another avenue to explore once you find out what your monthly budget can support might be to contact Habitat for Humanity. I believe they are mostly focused on families with children and you don't say whether you are in that situation or not. They also require hours of "sweat equity" where you help in the building of your own house. Having a requirement for the size of house you want and parameters like a finished basement on a budget is going to narrow your choices down quite a bit.
Once you talk with lenders to get a general idea of what they could lend you on a home, explore on Zillow or Realtor.com to get a ballpark idea of what is available within your budget. You don't say if you're also wanting to live in a specific state or area, so the more flexibility you can have, the more options you can find. There are some areas of the country where 1200 sq ft houses with finished basements are more affordable than others, so the more open you are on looking outside your exact specifications, the more options you might find.
If your monthly budget is very limited in addition to your down payment, explore what opportunities there might be for first time buyers. There is not just the down payment to consider; once you purchase a house, you also have monthly mortgage payments, insurance, taxes and repairs and upkeep to budget for, too. Knowing those things up front can help you be realistic about your search. It might take you doing some research, talking to social service agencies or even possibly realtors, for referrals. Another suggestion I would make is to not max out your budget to where all your money is going to your house. There will always be unexpected things that happen in life where you're going to need to have enough money to take care of something you couldn't anticipate. Lenders like to make you think you can afford "bigger and better" but you have to know what you can comfortably afford and stick with it or you could not only end up being unhappy but if you had a real financial emergency somewhere down the road, you could end up putting your housing in jeopardy. If I had maxed out when I got my house, when I lost the job I had I could have ended up losing my house by not being able to keep up with the payments on my new, unexpectedly much smaller income.
I purchased my house with a small down payment by changing my expectations about what I could afford and what area I wanted to live in. When I saw what was available within my budget in the area I wanted would require a lot more fixing up than I was able or financially prepared to do, I realized that my original expectations would be very hard (like, impossible) to meet on my budget. Once I changed my expectations I was able to see other options and found I had a range of affordable, realistic choices in a different area than where I stared my search and I've been very satisfied with the house I picked out from several in a different area from my original ideas for the past 10 years.
Do your research, lay the groundwork and enjoy your search.