I think I annoyed my previous T, but I wasn't allowed to email, only phone calls, and I HATED calling. I can probably count on one hand the number of times I needed to call but every time she called back I thought she sounded annoyed. I don't know if she really was or not. She always was OK with it next time we met.
"New" T is so stinking kind and sweet that she actually thanks me for emailing her every time, and tells me to stop being sorry for bothering her. She says that emailing her how I'm doing and what I'm feeling, expressing my self in a way that I can't when I'm with her, is very helpful and good. So it's a good way to aid in the therapeutic process, according to T, that is.
I still try really, really hard to NOT send an email unless I just can't stop the thoughts on my own.
I think it's totally normal to worry about annoying your T.