Yes, both VT-31 and 35 conduct USAF advanced for the C-130. If he is AD he could come here or go to T-1s but the T-1s are few and far between. The Guard guys seem to take those slots first. We don't get Guard down here I believe. I ask the studs when I remember to and in 2.5 years at VT-31 not one was a Guard bum.
VT-31 has a pecking order for the T-44C (glass) depending on fleet platform (i.e. USMC C-130Js) so a random USAF dude may not get the glass if loading becomes a factor.
Studs seem to want VT-35 so they can get the useless TC-12 type rating.
Having flown both planes recently and the T-1 back in the day, it is a toss up. Each plane and squadron has its pros and cons.