You can pull from either tube, but you have to pinch off the other one with a pair of forceps.
The problem as you probably know is that the pigment settles just about everywhere if it isn't kept suspended some how. The circulation pumps keep the ink in the ink lines suspended, and the daily cleaning should keep the heads and dampers cleared out. So there are two likely problems that you face - the ink in the cartridge has settled out and you're just getting solvent and additives without the pigment up to the head and it's just printing a clear that you can't see, or the damper and/or print head is clogged.
Syringing from below the cap top may or may not help as you're pulling from all channels - you can't just pull white ink that way. But you should be able to see white ink in the cap top after doing this if you've pulled it up to the top of the damper. If you don't see white, then I would start by replacing the damper, then the head if you still get no white through the head. You could try soaking the heads in cleaning fluid and running a couple of strong cleanings as well.
This is assuming that you've already replaced the cap tops - if not, that would be the first thing to do.
If you don't use white much at all, I would recommend that you replace your white ink cartridge with a ESL4 Cleaning Cartridge. If you need to use white, put a fresh white ink cartridge in and perform a couple of powerful cleanings to prime the white lines. Then when you're done, just put the cleaning cartridges back in till the next time. No need to prime the lines as normal cleanings should get the cleaning fluid through long before the white has a chance to cause any problems.
Good luck.
