There are plenty of 27k - 30k options in the last few days of May. But there are also plenty of flights that you can buy for under $300. It's not exactly cheap but I would not redeem miles for under a penny.
As others have mentioned, only saver space is made available to partners. There is no point checking entry single one; if there's no saver space there's no saver space.
I think that this is a poor time to try to use your miles. Either pay up to fly AA or consider other, cheaper options. Frontier can get you there for $72 on the 28th (Saturday). This is an unbundled fare; you will have to pay extra to even take a carry on, but even after the fees you are still going to be paying less than half the price of AA. It may not be the best flying experience (I've never flown them so I can't say, but I'm guessing that you get what you pay for) but it will get you there cheaply.
Personally I am not a fan of ULCCs but if you are going to try one, the time to do it is when the fare is 1/4 as much.