Emmett says he's considered or talked to nearly 100 candidates who have expressed interest in the vacant seat.
"I've had a lot of individual one-on-one meetings, and I've obviously been engaged in a lot of phone calls from people wanting to express their opinion and from people who I have sought out to say, 'Hey, what do you think?' And so my vocal cords are getting a little tired and my ears are getting a little tired."
Eversole resigned last week from the seat he's held for 20 years and will formally step down on Saturday. He's faces a retrial next month on corruption charges. The resignation is thought to be part of a possible plea deal with the government.
Emmett says the political calendar does play into his decision.
"If I appoint someone who intends to run for the seat, then they have to start running, because the primary is next March and the filing starts in November, so that political calculation — which we all wish wasn't a big part — does have to be a big part of it."
Emmett says he's looking for certain qualities in Eversole's replacement.
"One of my biggest factors is a maturity of judgment. I have trouble tolerating a lot of the behind the scenes type politics that goes on. So I want someone who can stand up, manage a big staff, manage a big budget and do the job for the residents of Precinct Four."
Precinct Four is a strongly Republican part of Harris County and Emmett is expected to replace Eversole with a fellow Republican.