BUT you are already sacrificing offensive potential to use a shield (no 2 handed weapon, no free hand to do stuff, no secondary weapon...)
You don’t seem to lose much by not wielding a secondary weapon, at least not without Feat investment. It doesn’t grant you any extra attacks, and if you want the extra accuracy that comes with using an agile weapon, you can just use an agile weapon with your main hand. It seems like the onliy Advantage to dual wielding is to get a bigger damage die on your first attack. With a two handed weapon you do gain some increased damage potential compared to sword and shield, but not a ton.
For the sake of DPR analysis, let’s assume you hit on a 7 with your first attack (and crit on a 17). 50% chance of a hit and a 20% chance of a critical hit on the first attack. The second attack increases your target number to 12 for a hit, a crit is no longer possible, and you introduce the possibility of a fumble on a 2 or lower. That’s a 45% chance to hit and a 5% chance to fumble, though since a fumble would be a miss anyway and fumbles on attack rolls don’t do anything unless the target has a reaction to take advantage of it, we can discount that possibility from our damage analysis. Your third attack, should you make it, has a target number of 17, giving you only a 20% chance to hit.
If we assume that greatswords still do 2d6 on a hit, and that they do 4d6 on a crit, that gives us an average of 7 damage per hit and 14 on a crit. (7*.5)+(14*.2)=6.3 average damage on your first attack. 7*.45=3.15 on your second attack. 7*.2=1.4 damage on your third attack. So if you make all three attacks every round, you end up doing an expected 10.85 damage per round.
With a 1d8 longsword, we’re looking at 4.5 average damage on a hit and 9 on a crit. (4.5*.5)+(9*.2)=4.05 average damage on your first attack, and 4.5*.45=2.025 average damage on your second. If we assume you spend two actions attacking and one raising your shield, that’s 6.035 average damage per round, plus you increase you AC by 2 and reduce incoming damage by your shield’s hardness every round. 4.815 DPR for +2 AC and some damage reduction doesn’t seem too bad to me.
Just for fun, let’s calculate dual-wielding witn a d8 longsword and d6 shortsword as well. That gives us the same 4.05 average damage on the first attack as the sword and shield. The second attack has a 50% chance to hit (still no chance to crit) for 3.5 damage, giving us an average of 1.75. The third attack has a 30% chance to hit, so our average comes out to 1.05, for a total DPR of 6.8 if we go longsword > shortsword > shortsword every round. That’s not even a full 1 damage per round higher than sword and shield at that accuracy, and this gets more favorable for the sword and shield if the dual wielded ever has to do anything other than attack three times.
All of this of course before considering any Feats or weapon properties like Deadly that might skew the math in favor of a character who has invested in building to take advantage of their preferred fighting style.