As long as the structure you are building off of is solid then it
should be fine. Here in the Southeastern US there are plenty of houses with porches built that way. Because of hurricanes we use strapping to tie everything together. I've built plenty of them. We would put a ledger board on the existing structure to set our rafters on. The outer corners are 4x4's and a header on top, with more supporting 4x4's in between for longer spans.
Given that the structure is solid with no racking, and the roof is properly nailed off to avoid racking, you shouldn't need much or any diagonal bracing. The only caveat being if there is no foundation under the 4x4 posts. Even if you don't want to dig and set piers, it might be a good idea to dig a bit at the surface to remove any organic matter in the soil (if needed) so it doesn't sag over time. Fill, tamp down, and level off as needed. I wouldn't try this in any place prone to strong winds, but otherwise it should work. If you are also using 2x6's as sort of 'ladder rungs' for storing lumber, then that mitigates the bottom of the posts shifting towards or away from the building. Perhaps find something to keep the bottom of the post from shifting side to side such as driving a piece of rebar in to the ground. As long as they are kept from going side to side then there isn't anywhere for it to go but up. Once loaded down with
wood it would take tremendously strong wind to pull it up. It will be a fair bit of work, but when you need storage, nothing else can replace it. Good luck!