Ask Mr. Smarty Plants

Before we could figure out an answer to your question, we had to answer two other questions: (1) is there a San Diego,TX as indicated on your question?  (2) what is a Palo Blanco tree?

On Question No. 1, we found a town named San Diego in Duval and Jim Wells Co. pretty far down in the southern tip of Texas. Question No. 2, there are apparently about 4 different trees called "Palo Blanco" which means white stick. Since that is not the tree you are concerned about, we chose Acacia willardiana, if anyone else is curious.

While we were arbitrarily picking tree names, we looked in our Recommended Species section in South Texas for members of the Quercus (oak) genus that were native to that part of Texas. The one we chose is Quercus macrocarpa (Bur oak), which grows up to be a really big oak, so we hope yours will be okay. From our webpage on this tree:

"Conditions Comments: The species name macrocarpa, refers to the golf ball sized acorns of this tree. The leaves of bur oak also are large, so they are easy to rake. Bur oak is drought resistant, long-lived and reasonably fast-growing for an oak. Tolerates limey soils better than other oaks. Resistant to oak wilt and a number of other problems. Sensitive to root zone disturbance caused by construction."

All this is not to say that either identification we made is necessarily correct, we just needed to know what ballpark we are in. Our concern (as is yours) is solely your little oak. You really should take a look at Texas oak wilt.org to acquaint yourself with where and how this disease can strike. In that website is a map of Texas counties, Oak Wilt Occurences in Texas Counties, that indicates that Duval and Jim Wells Counties appear to be oak wilt-free, but are just one county south of where infection is known to exist. If it turns out you have a red oak or a live oak, you need to be even more concerned.

For now, the important thing is to save your tree and protect it from the possibility of infection. If any of the branches are actually broken, yes, they should be pruned, even though this is the wrong time of year. It would be better if you called in a professional arborist, but you should look at Tree Stewardship in the same article.