Managing 3rd party clients

I've joined an organisation that has a number of clients, some of which run waterfall due to the nature of their industry and in how work is signed off.
My organisation has adopted some agile principles where possible, but is still very much tied to time-boxed, back-to-back, right-to-left commitments for projects. This means that people are often overstretched as commitments have already been agreed (by business).