Where to find a slime chunk?

Solution 1:

Caves generate in roughly equal proportions in all biomes, but a big percentage of caves under the ocean generate flooded, and as result spawn-proof against almost all hostile mobs (except occasional drowned).

Regarding digging, I'd suggest you go to level 11, and dig out the large 3-tall area there (possibly starting from a small island) - a good chance to get diamonds and other resources more valuable than what spawns at y=39, in the process and the same chance for slimes. Also light up any non-flooded caves in the vicinity, as it will significantly increase spawn rates of slimes - both reducing wait time to identify a slime chunk you dug out and significantly increasing drop rates of your farm once you build it. 1 in 10 chunks on average is a slime chunk, so if your example "test area" is 5x5 you have 93% chance of finding a chunk there. Make sure to light it up too - slimes spawn in any light level, other mobs don't.

Also, digging it out using a Haste II beacon and Efficiency V pick is a breeze.

Note, if you're playing on a server, you may need to wait until all players are logged out or in other dimensions, because they will be occupying most of the mob cap and you're quite unlikely to see any slime spawns.