The Degreaser + Axtinguisher answer to "a soldier that knows you're coming" is to skip the ax and reflect their rockets! That's one of the worst situations for trying to get an ax kill — if you're anywhere close enough, likely as not you'll end up both dead from splash damage.
The ideal case for Degreaser + Axtinguisher is when the enemy does not know you are there. In this case, you close in, ignite, switch to axe, and strike while they're noticing they're on fire — no airblast. The airblast is only for when they've noticed you (or are about to be in a position to) and you want to disrupt their aim, but it pushes them away so you can't ax them as quickly. Airblast only when it seems useful, not because it's part of the "combo". You should time your ignition so that you reach melee range just as you are able to swing the axe.
I recommend practicing the minimal ignite/axe attack to the point where you can use it in those encounters when you're almost as surprised as they are. (Play a map with lots of corners to bump into people at.) This is where I get the most benefit from the technique. I use mouse-wheel bindings like Steve V.'s answer suggests, except for not using the click.
What do you do with your Degreaser when you haven't got the close range? My favorite solution to this problem is the Flare Gun, which, like the Axtinguisher, crits on burning players. This time, you do airblast immediately after ignition, even if there's no wall; ideally, your airblast will put them on a semi-predictable trajectory for your flare followup. After you fire a flare, switch back to Degreaser and repeat from the top — if the flare hits, then a little more flames or another flare and they're dead; if it missed, then you want to keep them rattled for the next try using your airblast.
While you're doing this you should be keeping about at the end of your flamethrower range and circling around them to keep their aim off you, of course. If they seem especially discombobulated, or are reloading, immediately go in for the axe. Conversely, if you tried the straight axe combo and missed, airblast them away and switch to flares.