Yes, moving ahead to the run through the Underdark sounds great.
The plan -- as I understand it -- is to bring the ice toads along at first, help them find a satisfying snack on the way (while they in turn help us to get past an obstacle), then ditch them so we can speed along. If no satisfying snack presents itself, and so if going with the ice toads would slow us down, then we could instead just point them in some promising direction while we press ahead.
I can't even remember the last time the party had a long rest. And we're unlikely to have one in Termalaine, at least not upon arrival. I get that our lack of rest is due to a series of choices we've made under pressure but, at some point, we will need to rest. Let's just get to Termalaine as quickly as we can, do the necessary recon, and decide from there.