If the old chain was badly worn, the freewheel is likewise toast. Search the forum for "new chain slips on old cassette" and you'll see this situation is classic. A new freewheel is the only solution.
As for the slack lower loop, some sag is normal, but if it sags more than 1/2" below the straight line level, that's too much. However it's probably not the spring, but that the Chain is long and the RD cage is hitting the stop. If it were the spring the chain would sag in all the gears to verying degrees.
__________________
FB
Chain-L site
An ounce of diagnosis is worth a pound of cure.
Just because I'm tired of arguing, doesn't mean you're right.
“One accurate measurement is worth a thousand expert opinions” - Adm Grace Murray Hopper - USN
WARNING, I'm from New York. Thin skinned people should maintain safe distance.