You cannot benefit from two shields at once. They both provide Shield bonuses to AC, which do not stack with one another. Moveover, as you note, those bonuses become void the moment you actually attack: rather than give a bonus to AC, the bucklers start giving you penalties to attack.
Your attack routine therefore looks like this:
Main: +1 (BAB) +3 (Str) −2 (TWF) −1 (Buckler) = +1
Offhand: +1 (BAB) +3 (Str) −2 (TWF) −1 (Buckler) = +1
AC: 10 + Armor +3 (Dex) +0 (Shield)
When not attacking, your AC is 10 + Armor + 3 (Dex) + 1 (Shield).
A +1 attack bonus is very bad, even at level 1. Do not expect to reliably hit much of anything with that attack bonus; a typical low-AC target is still looking at AC 14 or so. You’d only have a 64% chance to hit even once against that target. Meanwhile the barbarian’s looking at around a +7 attack bonus, for a 70% chance to hit, for a whole lot more damage. Your odds of hitting with both attacks, 16%, is so low that you might as well not even have Two-Weapon Fighting.
That said, you really do not need to take these penalties to play the character you want to play.
Armor in Pathfinder is fairly heavily abstracted; one type of armor can actually cover quite a lot of fairly-different sets of protective gear, all of which is simply abstracted into a single armor bonus, a single armor check penalty, a single maximum Dexterity bonus to AC, a single weight, a single cost, and so on. Thus, while one person’s chain shirt might be literally that, a shirt made from chain links, I see no reason why you couldn’t describe yours differently: lighter protection on the body, perhaps, but heavier arm bracers, used to deflect blows.
Thus, your description becomes just that: description. There is no need to fall into 3.x’s “you need the feat [item] for that” fallacy. Simply describe your character the way you wish to describe him or her, and use the items that most effectively represent that description mechanically.
I cannot vouch for how nit-picky your PFS DM will be on this, however. Some DMs are incredibly nit-picky about completely unnecessary things, and in PFS they have some institutional backing for that. I personally dislike the PFS immensely, largely for this reason. Normally, I’d say if a DM gave you a hard time about this, that is a DM you’re better off not playing with, but here you may not find one who will. Ultimately, you may not be able to effectively make the character you want to make, at least no without taking absolutely pointless penalties (as with the paired bucklers).
The Special Edition PHB disambiguates the wording. Some people refer to this as "stealth errata" since it doesn't seem to have made it into any other sources, nor was it conspicuously announced anywhere. Even the SRD doesn't seem to have been updated to contain the new wording. Nevertheless, it makes the intent of the feat clear. Emphasis mine:
When using the attack action with a melee weapon, you can split your move action
in that round in order to move both before and after the attack, provided that
your total distance moved is not greater than your speed. Moving in this way does not
provoke an attack of opportunity from the defender you attack, though it might provoke
attacks of opportunity from other creatures, if appropriate.
You can't use this feat if you are wearing heavy armor.
You must move at least 5 feet both before and after you make your attack
in order to utilize the benefits of Spring Attack.
Notably, this is still not the same as Pathfinder's implementation: instead of combining the attack and move into a single full-round action, it keeps them as two distinct actions but allows you to apply them in an unusual way (much like the Overrun writeup that you cite). But neither implementation allows you to make a spring attack using a single standard action.
If you want to houserule Spring Attack to take a single standard action, go right ahead. No one will stop you. I might consider such a thing myself in my game, if it ever comes up. But that's not RAW, and in one of those rare cases where RAI is demonstrable, it's not RAI.
Best Answer
No. You get an extra attack from the off hand, but it is not an exception time-wise to the general rule: