I really wish you were in a position to post a good macro photo of the ants. I'm going to take a guess you have Solenopsis wagneri plaguing you because you stated you are in Alabama.
Bait traps can be non-discriminatory in this particular situation and use of same could actually backfire on you. Barring that, baits are a pesticide and would only provide temporary relief on your property anyway. You really might want to consider checking into parasitic flies. Me personally, I'd nuke those horribly invasive exotic fire ants (Solenopsis wagneri syn. S. invicta) with Phorid flies. By now, you can probably contact your local extension office to ask where you might be able to purchase Phorid flies for release on your property.
It will only take a few minutes to read what is at the site posted below and please note what is stated about non-discriminatory pesticides being "thought to have helped spread fire ants even faster by wiping out pockets of native ant resistance to the RIFA invasion. It also produced a toxic and persistent byproduct, dioxin, which accumulated in ecological systems”
http://uts.cc.utexas.edu/~gilbert/research/fireants/faqans.html#chemical
Hopefully after reading the above you have a big fat smile on your face because hope is on the horizon. I whole heartedly believe Phorid flies may ultimately be our salvation in fighting off these fire ants or at least reducing their numbers to that which would be manageable. Fight fire with fire. No chemicals involved in the process. Phorid flies would pose absolutely no harm to your dog or your bunnies or to you or your husband.
Isn’t Pseudacteon tricuspis a wonderful little parasitic fly? I think in time many folk down south will be singing praise of the Pseudacteon tricuspis. It is already being introduced in the US. There's another even more promising biological control on the horizon which I personally think is pretty wild and has the greatest potential in the long run to help control and manage these imported fire ants. For some really exciting reading, do a search of Strepsiptera. I think they're working with it at Oxford so toss that word in your search engine too. Pretty wild reading but basically the male Strepsiptera works its way into the host which would be the fire ant and encapsulates itself which renders the fire ant sterile. What a pity. Shall I pass around a hanky to dab our tears?
Somebody at Texas A & M had a phenomenal sense of humor when he wrote this, "Do I care? No," Johnston said. "Once they're parasitized, they're sterile anyway...they're out of the picture," he said. "In fact, I'd love for the parasite to extend the life span because those drones are a drain on the colony. If drones, or fire ant males, would stay in the colony and suck up nutrients for another two years, it would be wonderful. It's the same thing with the workers: once they're parasitized, they don't sting, they don't contribute, they just drain the colony."
And here's a link to the wonderful little parasitic fly previously mentioned-
http://www.nysaes.cornell.edu/ent/biocontrol/parasitoids/pseudacteon.html
They (Pseudacteon tricuspis) are going to take time to colonize to afford relief. Biological controls aren't normally quick fixes.
Phorid flies, hopefully coming to a neighborhood near you soon-
http://www.encyclopedia.com/doc/1G1-161077172.html