I have both of these problems! It's taken me 20 months of weekly sessions to come close to trusting my counsellor, but I still find it hard to open up and when I do, I'm convinced that I haven't explained myself properly.
My counsellor clarifies things with me though, as he knows I worry about being understood, and he reads between the lines more often than not, so they generally understand what you mean, even if you weren't clear.
Since you'll see them again, then you'll have time to reword anything.
I just let it all out, all the stuff I'm too embarrassed to tell others, its a relief, I don't think you can tell them too much, thats what their there for.
I feel more relaxed about it now and yeah i do clarify stuff if i think i worded it wrong. Now i have been doing it for a while thinking i have said to much isn't something i usually worry about and sometimes i do find myself wishing i had said more about some stuff.
I had been thinking of stopping though but i am unsure if i want to.