I'd say that depends of the weight of your roof
in that case I'd recommend x-bracing on all sides that don't have t-1-11. Not for every space, but on every corner, both directions. When I think about it if you're going 4' O.C. with your 4x4's, maybe a type of V-bracing would work, but that always looks incomplete
this is one way, although I've used a modified version before. The best angle is 45 degrees although I suppose you could go down to 30 degrees and still have an impact. I'd want to go from the deck all the way up and tie into your beams. But yeah, that's going to look pretty industrial and you probably couldn't get approval from your war department
it's prettier to do something like this (imagine a roof instead of a deck):
but to clean it up you'd use this modification:
notice how the brace is the same dimension as the post & beam. That would mean you'd use 4X4 braces, cut on 45 degrees and then lag bolt the braces to posts and beams. That takes some time to get clean cuts and to cut/drill mortises for the lags to rest on, but it looks the best
also notice the blocking in between each rafter. You should put blocking on top of each beam in your deck, and on top of each beam in your roof. You should also use some type of Simpson hurricane clip to tie each rafter to your beam. They are easy to install (after blocking), and there's the bonus that at least once you'll hammer your thumb (when you do, remember what Mark Twain said: "
profanity offers a comfort not available in prayer")
by the way, I don't think adding an extra post to each side is going to do much for wobble. I probably wouldn't bother
I don't know what type of roof you're planning on. There are shed roofs, gable roofs, hip roofs, etc. Now, a shed roof is easiest to frame
but you'll notice that one end is taller than the other. So, you could have 8' posts on one end and 11' posts on the other...welcome to wobble city. Now, you mentioned T-1-11 on one end so if you had that sheath on the tall end you'd mitigate that problem
if you're going to use composition shingles, the minimum slope is 2/12. That's 2 inches of rise for every 12 inches of run. So, for a 12' roof, if your low-side posts were 8', your high side posts would be 10'.
you can also consider a metal roof with a baked enamel finish. Comes in lots of colors, is much lighter than comp, is much easier to install, and can be applied to less of a slope than 2/12. Plan on bleeding though. I always do whenever I work with metal. Now, if you go that route it's important your finish roof dimension matches the width increment of your metal roofing. 1 foot roof panels can be attractive and easy to install. But that means you want your roof frame to be 14' or 15' or 16', not 14'5"
a gable roof is a little more complicated:
notice that there is a ridge rafter that runs the length of the roof. You'll want one of those. And it should be 0ne dimension larger than your rafters. In other words, if you're using 2x6 rafters, the ridge should be 2x8. Which brings up what you support that ridge rafter on. I'd recommend running your roof beams on all 4 sides. If the 16' run is your bearing beams for the rafters, than you'd use inside 4X hangers to install the two 12' beams (11'5"). Two advantages to that (and I'd install them in the shed roof too). One is that you can support the ridge rafter with a short stud on top of the beam. The other is this allows you options to add bracing to mitigate wobble, if it's too much when you get to that point
also, notice the collar ties/ceiling joists in that frame. You should put at least one at the mid point
if your wife wants a hip roof, you should probably just shoot yourself