Have you considered the shaking hands might actually be caffeine withdrawal?

Certainly it is not the case. It occurs specifically when drinking tea. I can spend a day with only my morning coffee.

>When I drink tea, I have shaking hands, especially with the bottled ice tea from the supermarket

Could be the tons of sugar? Or other artificial crap?

>Let's say I wanted to change from coffee to another hot but not sweet beverage: which one?

Tea is great and has health benefits.

The tea in "bottled ice tea from the supermarket" is not representative.

I can recommend green tea. It has also a waking effect but not as strong as coffe and in my experience longer lasting. It is more subtle.

There are many green teas with different tastes and you have to infuse them the right way. Worth noting is that you should not use teabag green tea but loose one. The teabag tea usually has a bad quality.

If you infuse green tea you should not do it too long (usually about 1,5min) and the water should be lower than 80°C. You need to experiment there a bit to find your perfect timings.

TLDR Green tea is worth a try, just don't use a cheap one (avoid teabags).

Sounds a lot like what my dad is drinking :) so it will be easy to try next time I see him.