I'm a ripper for large pieces as well. Particularly with backing - if you know how it needs to be cut/ripped (that's always my challenge). I've not found extended damage when ripping. I press the edges and then trim as another poster mentioned. I've never had an instance of the fabric being damaged beyond the initial tear which always works out after ironing/pressing.
If ripping is not for you, I would suggest finding at least close to center of the length of your fabric and fold selvage to selvage. If you see a 'bubble' at the folded edge, wiggle the selvage edges until the bubble disappears. Pin to hold that position. I then lay that section on my cutting table and smooth out to the far edges from there. Eventually the selvage edges with 'wander' and you will have uneven edges but your fabric should be very close to on-grain at that point. Trim edges and go from there.