As a fellow LGBT, I can tell you that this is not at all legal, as Federal Law makes it totally and completely illegal...The problem isn't that it's not illegal, as it is. The problem is proving that the reason your landlord is evicting you is strictly because you are gay and they found out about it...What reason did your landlord give you for evicting you? I'm sure you came home one day and out of nowhere found an eviction-notice hanging on your door and stating the time that you had to get out, etc. But when you spoke to your landlord after getting the eviction-notice, what was the reason he gave you? If you can produce cancelled-checks or receipts showing that you've paid every month's rent in-full since you moved-in to the apartment/house, and you've been given no other "warnings" or issued no other written violations of your lease, etc., then you can certainly fight this...
And I'm not suggesting that you guys stay there long-term, however it's going to be extremely difficult to find a home/homes for all of your birds, especially a temporary home for the two Green Cheeks, as it's going to be open-ended, the people will have no idea how long the birds will be there, etc. So what I'm saying is that you should call a lawyer and at the very least have the lawyer file a "Wrongful-Eviction" suit in your local court. You don't necessarily have to go-through with the suit, you just need to file one in-order to halt the eviction. The court system is extremely slow, and a judge will have to automatically file an injunction blocking the eviction from being carried-out until you go through all of the hearings, etc., which you can keep waiving and asking for extensions/postponements, etc. until you either find a new place to rent that will allow you to keep your birds, or until you find a temporary home for all of them. Then you can drop the Wrongful-Eviction suit all-together. It won't cost you anything to go and speak to an attorney who specializes in housing/leasing laws, the consultation would be free, and he would be immediately able to tell you whether or not you have a valid Wrongful-Eviction suit or not, and how much it would cost you to simply have him "file" the suit, which will automatically stop the eviction and thus buy you some time...And honestly, as long as you've not done something that makes it legal for him to suddenly evict you, then you've got a case...However, if you've even paid one month late, or you've violated any of the terms of your lease at all, even a little tiny one like you put a nail in the wall when you weren't supposed to, then he's got you. Also, if your lease is up for renewal then he has the right to ask you to leave...But if it's an actual "Eviction" notice that you got, and he didn't give you a legal reason for the Eviction, then this is exactly what you need to do in order to literally buy yourself some time so that you aren't forced to give-up your birds.
The court system is there for a reason, and it's extremely slow. This is one of those emergent situations where you should take advantage of this and use it to buy some time...I hope you are able to find a place quickly, I wouldn't think that finding a place that allows birds would be too difficult, dogs yes, but birds that live in cages, and they're all small birds at that, shouldn't be too difficult...you might want to look at private rentals that are being rented "by-the-owner", like small houses or duplexes, as a private individual will be much more likely to allow you to keep your birds than a housing-complex owned by a company/real-estate agency will...
Good luck, I wish I was closer to you because I'd certainly help in any way I could. But I'm in central Pennsylvania and only have one close friend who lives in TN, in Knoxville, and he's not a bird or animal person at all anyway...