If you live in Travis County, Travis County Sheriff Greg Hamilton's office will sign any Form 4 or Form 1 as long as you pass their local background check.
The Austin PD and Travis County District Attourneys Office (Ronnie Earl) will refer you to the Travis County Sheriff (I have asked both in the past) as the Sheriffs office keeps the "files" on all NFA owners and can recite to you every form they have ever signed off on. (They have asked me about specifics of previous transfers during a future request).
I can personally vouch that Travis will sign for Short Barreled Shotguns, Short Barreled Rifles, Supressors, and Machineguns. I have never tested the waters regarding AOW or large bore Destructive Devices but would assume that by default AOWs would be OK.
Williamson County used to sign for everything as of 6 to 9 months ago. There was a change in policy and the last I heard from my C3 dealer as of a month ago (who is in Williamson) was that he had a sit down conversation with the Sheriff and the political liablity/fallout was too high to continue signing. However, this may just be for suppressors as I didnt go into great detail as it didnt really effect me. However, if you want to know the specifics of Williamson give Chuck Tooke a call at Chuck's Class 3 and I am sure he could fill you in.
If you live in the City of Round Rock, the RR PD will sign for most anything.
James
Austin, Texas