You can probably take your shaman through the LFD up to 3 at a time, I would take the healer and tank through solo.

Once you are high enough iLevel to enter LFR you can have your team geared enough to do box dungeons in two weeks. Actually 2 days if you use the LFR on Monday night then Tuesday morning.

Here is a trick for super quick LFR (and maybe LFD) queues for your dps. Bring your healer then just drop the healer out right away once you enter the dungeon. Kindof cheating I know.