I agree with Outdoor Girl that this is an opportunity for you to make sure they know how important it is to get someone to take over your OLD job. So that you can concentrate on your new job, so you won't make mistakes.
Or to get their assurances that it's OK if you switch to doing your old job for 2 hours, and the new one for 5.
Love WillyNilly's advice as well.
As for how to not burst into tears of frustration, etc. I'm not sure.
But maybe go in with the mindset that these are your senior partners, and you are asking for their help, and getting their feedback, and you will be giving yours.
Also, do think about the mistake without panic. Is there a way it can be avoided? Is there a procedure, a checklist, a filing system, a communications pattern, a form, that can greatly reduce the chances it will happen again? What did contribute to it--did you not have enough time to "research" the task/person? See if you can identify causes.
It could be that they're aware, even if only subliminally, that this complicated and important new position is only getting a small fraction of your energy & time. And that has created a background feeling of "you're not doing your job right." or "things are out of control."
So you can say, "I need to give you the confidence that I'm focused on my new job, and I can't do that unless it's true. Which means I need to offload my old tasks."
One thing about moving up: lower-level work is absorbing and seems important. It's easier to do, often. But it's actually less important. Efficiency experts often warn against getting sucked into the clearly defined, urgent-seeming tasks that actually are less important that the more complex, more thought-involving, more long-term tasks on the higher level.
Mention that--you need to be freed up.