Have you ever struggled with finding diamonds? Here are some tips that might help you.
#1: Diamond ore spawns in layers 16-2. To track which layer of the map you're on, press F3 and look for the letter Y. It tracks your altitude, so as you move up, the number next to it goes up and vice-versa. The recommended layer to find diamonds is layer 12 (along with any other ore).
#2: Veins of diamond ore will spawn at least 3.097 times per chunk, or possibly more. The amount of diamond ore  in a single vein will vary between 2-8 blocks, or more if multiple veins spawn together, but less if an underground structure intersects it (dungeons abandoned mine shafts, strongholds,etc.). It is also possible (but rare) to find a vein of diamond ore linked with another vein of ore.
#3: To find exposed diamond ore, try to empty out lava pools for more surface area to search, or (as you already know) in a cavern, or abandoned mine shaft. Locating a mine shaft can be hard, but deserts almost always have one under them. Emptying lava pools are quite easy, along with locating them, but getting out the obsidian can be very time consuming, so to get the job done faster, try to just replace lava with another block.
#4: When you discover diamond ore, it is recommended to first mine around the mineral to ensure it doesn't fall into a cavern, lava, a dungeon,Herobrine's lair (trololololololololo),etc. Also (if possible) it is recommended to
have a fortune pick in your inventory to harvest more than one diamond per block.

Maybe it'll change in future world-resets, but the last time the Mining World reset, it seemed like all blocks were in the same place as last time.  If you're OCD like me, taking notes might help you re-find veins of ore and diamonds, after a world-reset.  I was able to find the same cave system, same abandoned mines, same zombie spawners, and the same veins of ore which I mined the previous month.  A few times stuff seemed to be off by one or two blocks, but that might have been me rounding the coordinate numbers up or down.