If you are connecting to Tor via AirVPN, this is not so. AirVPN would only be able to discern that you are using Tor, as it naturally follows that they would be able to see which guard node or bridge you are connecting to. They cannot see what is you are doing via Tor and therefore would have no record at all of your browsing history in Tor Browser etc etc.
There are several ways to use Tor with VPNs (for different use cases), but installing Tor Browser and running it after you've connected to AirVPN is the easiest way to get started (and sounds like it is exactly what you want). No changes to your configuration are needed.
(The Eddie client can also facilitate connections to AirVPN via Tor, but if you are trying to obscure your Tor use from your ISP this is definitely not what you want.)
Depending on what country you live in, if you've been searching for privacy related technologies from your home connection sans VPN then your interest has probably already been "flagged". (Consider: XKEYSCORE) But in any case it is true that heavy Tor usage might raise eyebrows even in locales where Tor use is not regulated. I think your caution is reasonable.