Contrary to some previous information, in Texas there is no law that requires you to be pre-approved before submitting an offer, although many sellers will ask for it to prove that you're actually able to afford what it is you're putting the offer in for. Still, it is not a requirement, just prudent to have one.
Obviously, your choice of location depends somewhat on where your husband will have to commute to. Presuming he's going to be downtown, the the choices you've made are all good, but each offers a different neighborhood feeling.
Are you looking for proximity primarily to a Catholic School, or very good private schools? There are some very, very good private schools that are non-denominational - I'm Catholic by upbringing, so I don't mean to slight anyone.
Price will be your controlling factor based on the neighborhoods you indicated as acceptable, and what square footage you're willing to look for. Tarrytown will run on average $300/SF, Clarksville/Old Enfield - $350/SF, Enfield closer to the lake (further West) - $325/SF, West Lake/West Lake Hills - $320/SF.
Note the pricing per square foot does not reflect the condition, as closer into town you tend to get high cost per square foot, but not necessarily as good of condition because your paying much more for the land.
Let me know if I can be of service,