You would then add your DEX modifier, right?

No.  The spell just means that when you work out AC whatever the result, if that result is below 16, it becomes 16.

Barkskin does not work together with anything: here is the Sage Advice ruling regarding Barkskin (which was a reversal of a prior statement on using it with equipped shields) -

How does barkskin work with shields, cover, and other modifiers to AC?

Barkskin specifies that your AC can’t be lower than 16 while you are affected by the spell. This means you effectively ignore any modifiers to your AC—including your Dexterity modifier, your armor, a shield, and cover—unless your AC is higher than 16. For example, if your AC is normally 14, it’s 16 while barkskin is on you. If your AC is 15 and you have half cover, your AC is 17; barkskin isn’t relevant in this case, because your AC is now higher than 16.

Since Barkskin is 2nd level, one hour duration, and $#@!*&% CONCENTRATION, I suggest everyone consider using Mage Armor instead, perhaps by asking nicely the Sorcerer or Wizard in your party. In exchange you'll get a 1st level spell, EIGHT hour duration, no concentration; and you can use mage armor with a shield which would bring it up to 15+DexB.

"I fed an AI all of the most nerfed spells in the whole history of D&D and this is what it came up with."