All tarps and tents stretch a little at first as the stitching eases into its natural stretched position.
And all nylon tarps and tents stretch as the nylon absorbs water - which happens overnight due to your own insensible perspiration, dew (whether you notice it or not), and/or precipitation. This is a permanent feature of nylon shelters. Would that manufacturers provided lightweight polyester material. Polyester does not stretch, and some good tents have been made of it. The up side of nylon's stretch is that nylon's strength is in the stretch; it absorbs wind shock and other abrupt stress loadings.
Yes, elastic guy lines will keep your tarp from sagging. I make loops of 12 inches of round waist cord elastic through the pull-outs, and attach the guy lines there with taut line hitches (prussic knots) so they are easy to reach and adjust without getting out from under the tarp. The idea is to tension the elastic about half way so it has enough pull to keep the tarp taut through the night as it absorbs water while leaving some spare stretch to absorb wind shock or the incautious step when I venture out in the middle of the night.