Can we have an overall picture showing the problem?

yup, tried my best here

Heres a pic I found from last year with the previous sail I purchased. It was too big and ended up sagging too much... but it's how I got to the idea of giving the new sail some slack, it seemed to work well before it stretched, I'm just not too aware of the implications with having a sail be looser.

One of the shade sail companies we've used explicitly stated that 4x4 or 6x6 posts are insufficient, and they suggest telephone poles or steel pipe. We've used 4" diameter galvanized pipe embedded 4 ft in the ground, with concrete, and then built a "box" around each post with trim lumber to hide the post. Not sure I see a solution here without taking the whole pergola structure down, although a structural engineer may see something I don't. I'd worry about leaving too much slack in the sail as they're made to be pulled taut. If it's loose, it's prone to flapping and in a strong sustained wind, may start tearing itself apart.

I agree. This will be challenging because there's no way to support the center of the span of a long beam connecting the posts nearest to the house. I can vouch for the shade sails stressing the posts. I have a small triangular one supported by 6x6 posts. The posts themselves are strong enough but they're not set in the ground well enough to withstand the forces. It's a temporary set-up, but looks like it will be more temporary than I planned on. :-)

Thanks guys, appreciate the input!

Have you considered a truss? A truss can span 24’. A local truss company or lumber yard should be able to help you. Also, do you need a scissor truss to clear the small roof?