I just end up flailing and relying entirely on auto fire.
That sounds about right! Generally speaking, there are two ways to go with omni shielded ships: put everything on autofire, or spend most of your time treating it as a front shield that you can, in an emergency, rotate to cover other directions (at which point, well, it's an emergency - the fact that you aren't getting useful weapons fire at the same time isn't relevant.)
I'll also note that, especially for larger ships, it's often faster to turn your shield off and then back on in a different direction than to wait for it to rotate.
For an example: the Aurora has a 180 omni shield. If I don't have a front shield conversion installed, I generally play as if that were a 180 front shield, using the normal rotate-ship-to-point-at-cursor gameplay, and only adjusting shield angle in special cases. Back when the wolf had an omni shield, I'd do the same, only adjusting it when there were salamanders incoming.
By contrast, ships like the Conquest, I tend to worry about navigation & shield positioning, and leave weapons control down to "what enemy ship do I have targeted" and "which weapon groups have I turned on or off".