So, you started experiencing these overwhelming emotions *10 years ago* (?) and right now you're not able to maintain the level of well-being you want, not able to manage those emotions unless you contact her regularly out of session? Sorry if that's sounds harsh: I realize I don't know what your starting point was in terms of if you had other issues to deal with, self-harm or other disorders or such, but 10 years is a long time to be mired.

Admittedly, I'm not a patient person.
I would suggest some serious adjustments are necessary. I'd consider adding sessions, doing more intense work around this, or changing therapists.
I don't think the issue is not relying on someone: we all do best when we have a support network, but if it's been 10 years and you don't have enough distress tolerance skills and external support to not expect prompt, thorough replies from T... it's just worth looking at.
As for payment, I see how it could be a touchy subject. I would approach it as: "I really respect your time and don't want our relationship to feel unbalanced because you're spending out of session time helping me without any reciprocation. Can we add one paid email to our therapy schedule per week?"
Or something like that. Perhaps it can be a less expensive alternative to another session and give you that written communication to hold onto in between live work, at least til you manage the DBT skills and make the larger adjustments.
P.S. As far as "trying to shake the neediness" I would not try to shake it. Feelings you try to push away don't stay away. I'd sit there and feel the unbearable-seeming neediness. Let it eat me up and accept it. When I do that type of thing, I often find as painful as it was, the power of it is defused and I see it was just another experience, I can put it in perspective, my life goes on. The feeling of "needing" anything besides food, water, oxygen, and shelter is really just a feeling of wanting something. Sure, there are things that enable us to thrive better, but neediness around contacting her is really just a want.
At any rate, I don't think wanting things is unreasonable, just a matter of whether your therapist can or will accommodate and whether or not it will help you in the long run.