Good luck with this.
Personal experience I did not like the AirTV. The interface was not very intuitive. The remote regularly disconnected and I could not get it to reconnect without a hard reset. Sling was not bad, but I live in a house with 5 bedrooms and tvs in use in most. Sling was too limited in the number of concurrent connections even when using over the air.
The worst problem of the over the air was when storms were in the area, reception was poor or non-existent. When severe weather is a possibility you will really appreciate the streaming capabilities of local channels. Having streaming capabilities over laptop, PC, phone, etc when power is out is another advantage to have local streaming stations.
While sling is available on other streaming devices, you do not have the local channels unless your tv is connected to a antennae.
Also we are more into sports, so that was another deciding factor.