I've had a particular amount of trouble against any morgonna, her shield absorbs my two attacks, neither of which will do damage to the shield and break it. And her hold comes off cooldown very often, preventing me from escaping when I need to. Is there any weaknesses I can take advantage of? or should I just avoid them and hope someone else on the team can kill them for me?

That's the problem right there, you're a support char, not a killing machine.

Actually, I played Morgana last night as Zilean and it wasn't that bad. Just be prepared to dodge her bind at all times and toss your time bombs on creeps to farm instead of on her to harrass. In team fights, she'll shield earlier and you can put it on whatever squishy she doesn't shield. It's also got a pretty low duration so you can always throw one time bomb on her to take her shield, wait for it to drop, then toss the other one.

When I played last night, the time bombs were actually exploding on her for damage when she shielded herself because the time bomb mechanic is basically to do damage once the time bomb debuff has been removed, the shield removes it, so she takes damage. I think that might be a bug though, and it didn't happen 100% of the time, the shield blocked the damage maybe 25% of the time, but I got her to kill herself with her shield a few times early game.

Just to throw it out there, Zilean can still put down some high damage if you go with an AP build (since it scales 1:1 with time bomb) and still maintain his support roll just fine. It actually helps you support by adding health to your ult. Just get a Shroud for survivability and cooldown and stack AP for the rest of you items and you'll be hard to deal with.