Would a more upright steerer tube be worse or better to alleviate this issue?

erm… What does flip floppy steering mean, exactly?

Do you mean something like a ‘lack of self-centring force’ when riding?

A steeper head angle would reduce it.

steeper = yes, higher = no

Tighten your headset right up, that’ll stop the movement

if its floppy floppy a shorterer stem may make it less flippish or floppish as it’ll quicker up the steeringness