Don't freak out just yet. There is an overlap of neon/cardinal temps (say 76F). It's not easy, but it can be done... I forget what other fish you want to keep...Tell us again please. A dwarf gourami as I recall...Is that right? Yep, you probably need to with all 1 species of tetras, but it's not an emergency.
Water changes are always good! Fish feel happy, & you lower any nitrate etc. There's almost no such thing as too many or too big WCs.
There is a bit of bossiness among tetras especially in a small tank with few members of a "school" of THE SAME species. You need to realize those tetras are used to being in huge schools of tens, hundreds or even thousands of fish...so when we suggest 5 or 6 of a single species, that is a very small minimum!!
It's not they can't get along despite any temp differences, it's that there are too few of any given species for them to act normally & be healthy. I thought you were on the right track with 11 "neons"...but that's really not true, is it ? ugh! the fish can tell they're not the same kind...even though we may have doubts, they know!!