When Rick Perry tried to encourage (or, according to the indictment against him, coerce) Rosemary Lehmberg, the DA of Travis County, to resign, he argued plausibly that her drinking problems, which landed her in jail, had eroded trust among Texans generally in the Public Integrity Unit she headed. So he threatened to veto funding for that office unless she resigned, and carried through on that threat when she didn’t.
Perry was probably right about the erosion of trust in Lehmberg, but who knows for sure? I doubt if Perry conducted any scientific polling on the issue, and Texas history is full of politicians who couldn’t hold their liquor yet served without controversy in their posts.
Lehmberg’s problems with alcohol and with the law would certainly have distracted her from her duties, for a while, but as far as I know Perry never offered any concrete evidence that Lehmberg’s office wasn’t performing its statutory functions in a competent way. It was all based on Perry’s notion, reasonable though it may have been, that there was a public perception of incompetence.
Perry’s own legal problems will certainly distract him from his duties as governor, for a while — probably far longer than Lehmberg’s DUI conviction and jail term distracted her — but Perry clearly has no intention of stepping aside as acting governor until his legal problems are resolved.
And if someone argued that being indicted for two felonies created a perception of incompetence on the governor’s part, requiring his resignation, he would hardly be moved to do so. And finally, if the Texas legislature voted to de-fund any of the various agencies Perry controls, in an attempt to coerce him into resigning, he would be well within his rights to take the legislators to court.
Texas has a law against attempting any such coercion of a public servant.