The AxeFx almost certainly has some kind of spectrum-dividing parametric EQ, right? Organs have basically NO treble signal, it's almost like the fundamental and just a couple lower harmonics - the "drawbars" on an organ select among those, but an organ has really discrete separation between the octaves selected. God for all the hype, the AxeFX thing better have some extremely competent parametric EQ stuff going... But typically when you kill treble, you sicken your mids too, and those, you need. And a "swell" function that plays with the attack envelope of notes - this is often found in with the noise gate functions, I guess there's a similarity in the processing that pairs those. And to be an organ, you'll have to track down a Leslie rotating speaker sound; and you should be able to lock that in tempo with a specific note time value, be it quarters or half notes or dotted eights or such. You want to do that because them you can sync a tremolo volume-wobbling function to it. EH made their "POG" boxes that separate out tone control for each part of the spectrum, and so many people were using it with a Leslie-type box to make organ noises they decided to put it all in the same box. If you think about it a bit, you can see that what makes guitars sound "guitar-y" largely has to do with the pick attack - you can back into organ noises with a volume pedal alone, but you have to EQ your signal into the organ range, and - play ORGAN parts. They don't tend to bend strings, for example....
They also make a "Ravish sitar" box - but to do that noise, all you have to do is have a delay that can be set as low as 3, 4, 5 milliseconds, then turn up the repeats and delay volume into the 90's somewhere - instant sitar.