Hi Butterflying,
The email boundaries with your T sound rather vague to me so I'm not surprised that it's hard for you. If you're not supposed to do it at all, but then your T responds once in awhile, he is giving you the message that it's not totally off-limits. Intermittent reinforcement is very powerful, so if your T really wants you to rely on internal coping methods he should set firm guidelines around email and stick to those limits by not responding at all.
I can send my T unlimited email to his office email address with the understanding that he may not have time to read everything and he rarely has time to respond unless I'm very clear about needing a response to a pressing issue. That works well for me because I'm not limited to a certain number per week, and he says he doesn't feel overwhelmed by them because we have an understanding that he may not be able to read all of them.
It's extremely important for a T to be very clear about those boundaries that are most problematic for a particular client/patient. While I can send unlimited email to his office address, I am not allowed to send any to his home email address. Last year I lost my email privileges for a month because I sent a couple of emails to his home address after being warned about that twice. Lesson learned, although I should have known better anyway because he's exceptionally consistent in following through -- he says what he means, and he means what he says. And I need that.
Personally, I think it's pretty natural to want your T when you feel upset or overwhelmed, and that's one of the reasons why my T allows unlimited email even though he rarely responds. He knows it helps me feel connected regardless of whether or not he responds, and because of my struggles with impulse control, being able to email him at work anytime reduces the frequency of other behaviors that he finds more difficult to tolerate, such as emailing him at home.
Maybe you and your T could come up with a more structured plan that will be helpful to you, rather than this vague set of guidelines that make it difficult for you to know if it really is or isn't ok.