does action affect string tension?

Raising the action won't change the tension. It might feel stiffer, because you'll have farther to go to the fretboard - but since you'll be stretching the string to get there, your fretted notes will go sharp.

The pitch of a string depends on three things: the string length (which you can't change without having to also move all the frets!), the string tension - which is the factor you want larger, and the string mass (gauge). So if you need more tension without raising the tuning above drop C, you'll need heavier strings.

, although i'm using heavy gauge strings, they still feel a bit floppy at drop C.

NoteBoat is correct; you need heavier strings. For Drop-C on a 4-string bass, you might want to try the bottom four strings from a 5-string set. As you'll be tuning up from BEAD, maybe a "light" 5-string set with a 120 B string would work for you. You'll probably need to reslot the nut though.

NoteBoat is correct; you need heavier strings. For Drop-C on a 4-string bass, you might want to try the bottom four strings from a 5-string set. As you'll be tuning up from BEAD, maybe a "light" 5-string set with a 120 B string would work for you. You'll probably need to reslot the nut though.

...and potentially re-adjust the action, intonation and relief. I agree with Gabba Gabba, you'll probably have the best luck with the lowest four strings from a 5-string set if you're tuning down to C.

Agreeing with all.

Best to go with the heavier strings, one more reason is that if you raise the action excessively high, outside of normal parameters, your intonation will be off when fretting notes. That's because the length of the string gets changed abnormally or excessively; a bit more (shortened) than the instrument is engineered for. You would then try to compensate at the saddles, it gets complicated, and I believe that the best you could do with intonation is to "get close".