As much as we hate the move process, I'd take whatever time is necessary to find a forever home & rent if I have to.
Since you've already listed, if you have a realtor, you have a contractual obligation to accept a reasonable offer. Selling now means you will have the money in hand & don't run the risk of owning 2 houses. Put the proceeds in a money market account.
Closing in June should allow you enough time to seriously de-clutter your possessions, assuming you can start right away or after a short option period (required in TX). You don't want to move anything you aren't going to keep. This process is time consuming, so don't delay -- we needed every day. As tempting as it is, don't house hunt until you have the de-cluttering done.
We're retired, decided to downsize & move to Austin, decided to build on a lake property there that we've owned for 19 years, worked with an architect to design an aging-in-place house, moved into a rental so we could demolish the existing old lake house, sold our big Dallas house, put the Dallas furnishings we plan to keep into storage, are busy making many selections, and can't wait to finally move into our forever home this fall! It's been a busy year but so liberating.