Year one exotics can be had from anywhere, they have a chance to drop from every mob or gear and weapon boxes, but the rates are low, targeted loot helps but not as much as before tu8. You can farm bosses for a higher chance, higher difficulty gives a higher percentage. Another source now for year one exotics with god rolls is the exotic chest from the season stuff.
You can also craft/quest for some of the year one exotics, namely nemesis sniper rifle, liberty pistol, dodge city holster, chatterbox smg.
Everything you need to get your hands on the Division 2 Chatterbox Exotic SMG and find all the parts
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Merciless rifle, sweet dreams shotgun, pestilence lmg, sawyer knee pads, btsu gloves, chameleon ar, are all random drops.
Eagle bearer ar is raid only drop, Diamondback rifle is kenly college drop for finishing it.
Year two exotics can be had in certain ways.
The bighorn ar from legendary levels, 5% drop rate.
Bullet king lmg from riker missions so wall street and tombs, 7% drop rate heroic.
Lady death smg from the named bosses in ny zones the deck of 22 bosses, 3% drop rate.
Acosta's go bag backpack random from any faction crates, 3% drop rate.
Tardigrade chest from true sons missions, 7% drop rate heroic.
Imperial dynasty holster from cleaner missions, 7% drop rate heroic.
Ninjabike kneepads from darkzone supply drops, 3% drop rate.
Coyote mask from season reward level 35 or from boss named coyote from jefferson trade center mission.
All invaded, heroic missions/strongholds apparently have 8% chance to drop all year one exotics, legendary levels have 10% drop rate.
Most of these are junk now since the talent rework, you can't recal, and stats are random just like everything else, there isn't anything there that are absolute required items, but they add a different flavor to builds, like liberty, acostas, coyote, tardigrade, and pestilence.