Spiders, Cave Spiders, and Cats all drop 0-2 string, meaning on average you should get 1 string per mob killed, and for every given mob you have a 66.67% chance for it to drop at least 1 string. During the daytime, you can use the pie chart to find the closest mineshaft for Cave Spiders, kill village Cats, or try to get lucky with cave spawn.

Striders always drop at least 2 string, and up to 5, so if you need 2 string quickly they could maybe be feasible (as long as the string doesn't fall in lava) but otherwise if you're in the Nether, Bastions are better.

The following table taken from the Wiki shows the respective chances and stack sizes that you can get string from various structures. Note that Mineshaft Minecart Chests do not have string in their loot table.

Cobwebs can be broken by Swords to always drop 1 string. This means the cobweb in Igloo Basements can give you 1 string, but more pertinent to Draftout are Mineshafts and Abandoned Villages. You can locate mineshafts using the pie chart, and if you have a bell on your pie chart, but no iron golem or villagers, you likely have an abandoned village (Taiga villages can spawn without golems so check also for zombie villagers if you think that's a possibility).

Stronghold libraries and Trial Chambers can also have cobwebs, depending on your line. Once I have a page for Obtain Cobweb I'll link it here. 

Piglin Bartering