I can also say that after quitting caffeine (8-9 months ago) my sleep improved dramatically for about 5 months. Then I started getting sloppy and mixed in light-caff (like chocolate and decaffeinated teas/coffees) and other factors in my life also kicked in and sleep got worse again. Unfortunately I don't have the drastic caffeine cut to fall back on now.
For me, the change 9 months ago was entirely one change - unsweetened caffeinated drinks to unsweetened purely non-caffeine drinks (like "dandy blend" and rooibos).
Anxiety didn't decrease noticeably when I made the change, but sleep improved a lot. Now if I drink caffeinated teas (3 times in the past 2 months) or eat a lot of chocolate I definitely feel the anxiety and just general discomfort. So there's a link of some sort. It feels uncomfortable now...