There are many ways to skin the cat. You just have to analyze all the facets and make sure they can handle the load.
But also keep in mind, different grades and species of lumber have an effect. For example, a pair of 2x12's spanning 10'....
If #1 SYP would carry 405plf. Given your 10' joists....that would equate to a 40.5 PSF total floor load.....
If #2 SYP You loose 100plf capacity, and thus only a 30 psf floor rating
If #3 SYP TAkes you all the way down to 180 PLF or 18 PSF floot. And even a 4-ply of #3's is on good for a 40 psf floor.
As to the exterior wall...yes, it only supports 5' of the floor.....so each foor of the carry beam only needs to be good to 250#.
AND, the beam is spanning a shorter distance. The result:
A pair of 2x12's at an 8' span if #2's is good to 474 PLF. Well over the 250 needed.
Even a pair of #3's over 8' is good for 284plf.
So yea, the exterior beam is good. BUT, at 250PLF, and 8' span......the post to beam connection has to support 2240#, OR 1120 per 2x12. So you would need about 11 nails each.
But what size are the footings since you used exterior posts designed to support the rest of the structure? Can the footing support an extra ton of weight? Footer sizing is usually based on soil type, and anticipated snow loading, etc.
As to the interior posts.....Yes if you put another post in, I would dig down to the footing. But I really dont think it would be that hard to notch the post. Remove the 2x12's one at a time. (a single 2x12 should support an empty floor above). Notch the post, put 2x12 back and toe nail joists. Repeat on other side.