Hello--don't know if this is a "new" Dimmitt or not, but I wanted to share what I found--
Alamo A. Dimmitt buried in the Texas State Cemetery
B. 1840 D. April 21, 1900

Section:Confederate Field, Section 2
Row:R Number:17

There are a couple of Dement surnames buried there as well.
http://www.cemetery.state.tx.us/pub/user_form22.asp

Alamo
Dimmitt
Died
Apr. 21, 1900
Aged 60 yrs.
Co. D. 4.
Tex. In.

Ella Hendrix
El Paso, TX
ehendrix1@elp.rr.com