I remember this happening years ago, maybe it was vanilla, where I’d click on someone to heal (no add ons) and then change targets by clicking on someone else but for whatever reason, the heal goes to the previous target instead of the current target. This is now happening in classic and I don’t know how to fix it.
Anyone else having this issue and/or know what I can do to correct it?
So you’re targeting someone, you cast a heal, but during the cast, you decide to target someone else who you want to heal, and you’re wondering why the first person you targeted is getting healed?
Just trying to confirm what your question is.
No. Here’s maybe a better example. I’m targeting player A and healing them but I see that player B is losing health. I then target player B and cast a heal on player B but instead player A gets the heal.
Does that make better sense?
Targeting is executed on key up, casting on key down. There’s a small window where some people can begin casting before the switch happens.
Someone else with the same issue:
https://eu.forums.blizzard.com/en/wow/t/spells-activate-on-previous-target-when-clicking-targets/96353
Easiest workaround is to use mouseover macros.
I’ve never had this problem but the above explanation makes sense. Just make sure you actually have your target acquired before you cast. 
Personally I often know who I’m going to heal next before the cast finishes on the first one, so I’m usually already on the next target by the time a cast finishes, so I’ve properly acquired it already and can immediately begin casting. It just sounds like you waited too long and are making split second decisions that are causing these targeting issues because now you’re in a rush to get a cast off before you can even finish clicking the target.
Realistically waiting a millisecond to properly lock your next target isn’t going to get anyone killed. This would be true for Retail but with 400ms spell batching in Classic, it’s even more true.
This has been driving me crazy. Literally have to wait a half second after switching targets to make sure the heal goes to the correct person. What happened to #nochanges? Priest healing sure wasn’t like this in original.
You don’t have to wait long. As explained in the post above, just make sure you wait to lift up your mouse button before you cast your heal. It should be a fairly quick action when you click the mouse.
The default in vanilla was to cast on key up. You can restore that playstyle by setting the ActionButtonUseKeyDown CVar to 0 (or options in your bar mods if you’re using non-default).
Alternatively you could buy a mouse with a shorter travel distance for clicks.