Saw this bird outside my apartment in College Station, Texas and have never seen anything like it before! It is about the size of a hand.
It is American Woodcock, Scolopax minor.
Superbly camouflaged against the leaf litter, the brown-mottled American Woodcock walks slowly along the forest floor, probing the soil with its long bill in search of earthworms. Unlike its coastal relatives, this plump little shorebird lives in young forests and shrubby old fields across eastern North America. Its cryptic plumage and low-profile behavior make it hard to find except in the springtime at dawn or dusk, when the males show off for females by giving loud, nasal peent calls and performing dazzling aerial displays.
The newborns are even more camouflaged in downs.
I've found whatbird.com to be pretty good for identifying American birds. Asking it for grey and brown medium-sized birds that are commonly found in Texas gives 14 options, of which the best match is the American woodcock, as suggested by Sanjukta (which is conveniently early on the alphabetical list!)