The only types of financing still available today that are no down payment are VA & USDA financing. VA requires you to be a Veteran, and USDA financing is for rural areas/towns (it is not available in Harahan, but it's available in Waggaman) + you cannot own a functional home which is in the same area as you are buying. With conventional financing you can purchase with as little as 3% down, and FHA financing you can purchase with as little as 3.5% down.
Since it sounds like you already own a home then by most down payment assistance programs you would not be a first time homebuyer and not eligible for down payment assistance programs (can't own a home in the past 3 years), so you would either have to come up with the down payment from your own funds or get a gift from family/relatives. Louisiana Housing Finance Agency the down payment assistance eligibility: http://www.lhfa.state.la.us/programs/homeownership/addpi.php
How much you can qualify for will depend on your DTI, like Jen said, however your DTI is not income compared to credit, is is the percentage of your monthly gross income your monthly debt payments are (car loan, credit cards, student loans, etc.) which would also factor in the proposed mortgage payment. For best chances of approval you'll want to keep that ratio to no more than 43-45%. For example if you have $3,000/mo in qualifying income, and with your debt payments + the new mortgage payment you have $1,250/mo, that would be a 41.66% DTI.
@ Jen Pfeffer - Milheath is in HaRahan Louisiana (suburb of New Orleans)... not HaNahan South Carolina (I was wondering why you mentioned Goose Creek).