Usually on Sym I act like a flanker and hunt the healers. I will make a death room and then once the enemies pass me I jump out to the back line to cook the healers and any of the squishier enemies with my max beam before the tank even realizes what happened. 90% of the time it works great, but now and then you get those healers that are just to good to harass that way (A mercy that flies between teammates every second, Lucio that uses proper crossfade swapping for speedy escapes, a very accurate sleep dart ana, etc), on those ones I just focus on pissing the whole team off. I did that to one team on Illios and they were so pissed that when they finally killed me they all teabagged me, even though they were losing terribly, lol. I ran back and murdered them in an overtime for the POTG.
Also don't hesitate putting shield generators in the defense starter spawn room. Whether attack or defense few people go into that room after the first cap and its so centralized it can cover a huge part of the map. The only one that is too risky is Route 66, since the spawn room is small and right on the payload track, thus often used as cover for both sides. You have to be sneaky about that since often times once the enemy knows you have a symmetra, a tracer or genji will go searching for the generator, and I find that room is usually either skipped or the last place they check.
BTW, Sym is now just under Lucio as my most played character, I simply loved her redesign.