TBH I've never know an overdrive sound that made me think *worship*. However I do recognise some overdrive sounds that make me want to leave the room, and the Hillsongs tone is one of them. In fairness, it's probably the style of playing rather than the actual tone that I dislike, and played differently I'd probably enjoy it a whole lot more. This isn't intended to be a rant about music particularly, other than so many worship tracks all have small variations of that sound and style that it's becoming hard to differentiate between artists, and it really detracts from the song for me.
For the little it's worth, when I play guitar - proper electric guitar, rather than synth or acoustic - then my overdrive comes from a Jekyll & Hyde (use the Hyde side for distortion - sounds nasty with Jekyll too) or Danelectro Transparent Overdrive or from my amp. Guitars are usually a Strat, Les Paul or Washburn A20V. Generally I want my overdrive to be a solid bar of tone that I can hammer and shape into a musical weapon that will make hairs stand up on necks.
Yes! I've been using the Fulldrive 2 Mosfet for over a year now. Huge fan of the pedal. I use a similar setting as you do with comp cut on but with tone a little past 12 oclock and with overdrive at around 10 oclock. I click on boost for solos and parts where I need to cut through the mix. It's a great "church" overdrive. :)
I think another great pedal for worship is the Jetter Red (squared or shift). You can get a really nicely textured / dark tone that has perfect character for expressive solos during worship sets.