Honestly I just holed up in the back left corner and then used a combination of grenades (I probably should have brought a few more medkits, but instead I brought extra grenades and only my standard "medic with 1 (3) medkits, and one person with a backup medkit in case the medic takes a bad hit") and rockets to annihilate pretty much all of the cover on the lower level, using a gunner to pin down any thin men that actually dropped on my side long enough for me to deal with them. My solution to acid (since I didn't have the medkits) was basically just to grab the edge so I always had at least half cover, then spread everyone out (so that only 1 could get hit with acid at a time) and just having whoever got hit repetitively take cover until it wore off. The fact that the thin men had basically no cover (since I grenaded/rocketed all the tombstones into oblivion whenever a thin man stood behind them) meant that the loss of 1 person for a couple of turns wasn't that big of a deal, and my backup medkit carrier was my best executioner infantry just in case I needed to pick off someone while she was acided.
Then I just slowly progressed forwards along that same side, killing anything that dropped on the lower levels with relative impunity. Hold out long enough and the thin men don't even really drop anymore, I must have killed like 40 of them over the course of the mission and when I finished there hadn't been a single one on the map for at least a couple of turns.