Even though you've recently played a paladin, would you consider a paladin multiclass different enough?
Your best bet to be anti-caster is to be an Abjurer or a bard, but you want damage too. Paladin/bard has multiclass synergy with the capability to grab most of what you want. Paladin 2/Valor bard X or paladin 6/Lore bard X will get the job done. With the bard's spell slots available, you won't have to worry about running out of steam. Save those Divine Smites for the right situations instead of using it indiscriminately. The Valor build gets better spell progression but the Lore build gets Charisma to saving throws, helping you to steamroll through a lot of common magic effects. You could also do paladin 2/sorcerer 1/Lore bard X to rely on Booming Blade/Green-Flame Blade instead of Extra Attack to scale your damage.
You want large individual hits to break concentration. Divine Smite and Sneak Attack are the two biggest, reliable ways to get 1 large hit.
Counterspell is helpful, but it's best on someone who will be able to invest a significant amount in their casting ability score with plenty of spell slots to spare. Abjurers and bards can get proficiency to such rolls to help (half in the bard's case). Keep in mind the order of operations on the Mage Slayer feat. The spell gets cast. It has to finish before you take your reaction because the wording is "cast a spell" not "attempt to cast a spell" or "start to cast a spell." If the effect of the spell requires concentration, the Mage Slayer reaction hits after the concentration is up, helping to ruin the spell. You only need to counter instantaneous effects if you're effective at sticking close to your mark.
Nondetection is pretty awesome. Nondetection beats all divination spells.
This is especially helpful if you can become invisible through some other source.
Nondetection lasts 8 hours without concentration, so you can also make yourself invisible and be undetectable by anything most casters could do. What is the rest of the party?
e: Arcane Trickster X/Abjurer or Diviner Y could accomplish what you want to get an AT base that also has enough spell slots for effective anti-magic. Improved Abjuration is a solid feature but takes awhile to come online. Portent can fix dispel/counter checks for you and is available quickly. You just have to find the sweet spot between just enough Sneak Attack and enough spellcasting power. Again GFB or BB can scale up your damage and both work with Sneak Attack.