Veneers are always the best option if there is enough existing tooth structure to attach the veneer to. If there is significant loss of enamel on the inside of the tooth, the dentist may recommend that you have it covered with porcelain and this would be a treatment that would be more like a crown than a traditional veneer.
You are right in that veneers are a better long term solution in both strength and aesthetics. Chairside veneers or bonding are when the dentist uses a composite resin and places it on and between your teeth, molds it to look natural and meet your desires. A very good cosmetic dentist can make chairside veneers, or bonding, look really great. It is very technique sensitive and the $ 600.00 per tooth would certainly be in the range of what a good dentist would charge for this. Lumineers are a brand name of porcelain veneers
Usually when a root canal treated tooth has sensitivity, the tooth may be a little too hard in certain movements. I would have your dentist evaluate your “bite” on this tooth and adjust as necessary. Have them take an x-ray to ensure that there isn’t something going on at the root of the tooth. On occasion, a tooth may have an extra root or chamber that may have been missed with the original root canal.
If the existing crown broke off at the gum line, a root canal needs to be done so that a post and core/build-up can be placed to establish a solid foundation for a new crown. We need something to attach the crown to and so a root canal and post provides this.
It could be that you have had some gum recession and the cold air is causing the sensitivity on the roots of the teeth that are normally covered by the gums. This does not necessarily mean there is a cavity or decay, just exposed surfaces of the roots. This can often times be resolved with a surface varnish or protectant over the roots or even a desensitizing toothpaste like Sensodyne. In extreme gum recession cases, a gum specialist (periodontist) may need to be involved and they can actually recover the root surfaces with more gum tissue.There is always a possibility that what you are experiencing is a micro gap that has formed between the veneer and the tooth but typically this would be the case if only one or a few of the veneers are sensitive, not all of them. Best advice would be to go in and see the dentist that placed the veneers to evaluate.