Should I tell her that I'll be leaving shortly?
No, you should just let things go as if nothing happened. After you have signed the new contract tell them that you are leaving.
Even more, as long as you do not have a contract signed, you cannot know when you will leave, shortly or not.
Otherwise, you risk that your future will not be very bright.
If they ask you why you did not tell them earlier, you just answer them that you did not know earlier, things just happened quickly. They cannot verify that.
And even if they ask for proof, you just tell them that they have to trust you.
On the other hand, if I'm promoted, I might be able to negotiate a higher pay with the new company.
One more reason to say nothing.
As explained elsewhere: when they will ask you why you leave, just tell that the new job suits better to your future development. Under no circumstances tell them the real reasons.