I drink decaf tea and coffee, but I'm sure that there's something else
that has an effect (tannin?), cause after about 4 or 5 cups of decaf
tea I still end up feeling a bit frayed. There's still some caffeine
in decaf, but it should not be enough to have a noticeable effect.
Not sure what could cause that.

Any recommendations for types of tea or coffee that would be
relatively free of side-effects? I've tried Celestial Seasonings tea
with Rooibos...pretty good in its own way, but not reminiscent of
black tea.

I would hazard a guess that placing hot liquids in your stomach excites
your system a little bit. Maybe see if you can handle iced tea without
similar effects?

The first part of the infusion process will release the caffeine in a
tea, and if brewed longer, tannins will begin to be released. Try
different brewing times, experiment with different tastes. A good
black tea should be brewed for three to five minutes, so at three
minutes, you should have yourself a good cuppa.