Actually, you CAN shoot and edit, also connect to the sound board for a feed - they do it in live studio productions all the time. You will need an AV mixer/switcher that allows multiple camera feed and audio inputs, monitors and communications for calling the shots, an assistant to help at the controls while you watch the camera angles on the monitors and tell your TD (technical director) what camera shot to go to. You also tell your camera people over your communications system to move in, move out, give more/less head room, lead the shot, focus, etc. and anticipate where you will be going with all this. At the same time you are rolling either tape or DVD recorder, capturing this on-the-fly stuff and essentially wind up with an edited mix at the end of the production.
Graphics, cutaways, titles, etc. are either added on the fly, having been prepared ahead of time, or if you have a really sophisticated system, in progress. And, with the stuff recorded on the fly and resulting (with time and experience) in a fairly decent edited production - you always have the option to do post production cleanup editing to polish it up some, if needed.
Of course, this can prove to be an expensive direction to take but, depending on the budget the first place I'd look would be at the available NewTek, Sony or Panasonic mixer/switcher systems. A LOT of houses of worship are incorporating just this kind of production hardware in their productions.