Echoing what others have said - use a HELOC. Can be closed quickly (14-21 days in some cases…). It is a second lien. No immediate need to withdrawal on it if you don't need to. Allows you to go make the offer on the home (assuming you can qualify if you need another mortgage).
Downside is you'll be paying close to 8-10% on interest. And in Texas, HELOCs only let you take up to 80% of the appraised value
There's other products though that can allow you to use the equity to buy a new home without having to sell your old home. Move into the new home. Clean and sell the old home and voila.
Contingent offers aren't bad though in today's market in most places. That's because you're likely not having many fish biting, so if the only fish a seller has that's biting is one with a contingent offer… they are likely to take it and at least consider. Again, very dependent upon the locale, but still.