Boy that is a great question and one that arguably has no simple easy answer.
And I just spent some time on your web site and your recordings are really pretty good , and certainly you have the basics very well covered (more on that later)
First perhaps just to make sure we are all on the same page the term "airiness" can mean different things, to some it means high frequency focus almost hiss but not ?? ( ya Kev like that makes a lick sense
)
To some it means air or space , perhaps more of a 3D feel , around the individual elements in the mix, which in turn can help the entire mix feel more 3D (for lack of a better term)
To some if means presence ( a more intimate right there in the room feeling also 3D ness ) particularly on vocals and sometimes guitar as well .
And yes some of these things can be achieved with judicious use of EQ and or compression ( often there in the mix but not noticeable until bypassed)
But in truth it can also be said that everything in the recording chain from player to instruments , to room, to mics, to pres, to AD conversion, can play a role
With all that wordy preamble here are some basic thoughts .
If you are referring to more presence , intimacy, right there in the room, and say more of 3 D feeling.
As far as EQ :
The one place to start to think about ,,,, is mud/distortion and build up of competing frequencies Particularly in multi instrument recording (i.e. anything that distracts or detracts, from clarity) and the rule of thumb for that is, get rid of mud first.
The best / first way, to have the individual elements become more detailed, distinguishable, and "present" as far as any EQ, ( And even though it may sound counterintuitive), is to first take away what you don't want to hear (as opposed to the usual tendency to add to what you do want to hear) Or in other terms "subtractive EQ" first before (if any) additive EQ is applied.
Also in multi instrument mixes do not try to have have everything front and center or forward in the mix all at the same time, or even at all sometimes . It is not only "OK" but often actually creates more 3D from front to back space and air, to intentionally have some elements further back in the sound all the time or at different times.
If you don't mind my asking what Vocal mic are you using?