Abilify or Geodon do not have antidepressant properties. The only antipsychotics with antidepressant properties are Seroquel (heavily sedating and appetite stimulating) and Latuda. Going for 'the next closest' is a stupid attitude to take by your pdoc, in my opinion. The 'next closest' thing to Latuda is Seroquel (but it is structurally very different).
Any antimanic will prevent depressions which follow mania.
*mod edit* Your antimanic does not need to be an antipsychotic. It could easily be something like Tegretol or Depakote.
From my very extensive (and awful) experience with antipsychotics (Risperdal, Abilify, Zyprexa, Seroquel), I'd say that all, including 'mild' ones (e.g. Abilify), should be avoided if a conventional mood stabiliser is available. The reason you have so many side effects is that you've been put on a class of drugs that seem to cause horrible side effects for the vast majority of people. Don't let antipsychotics put you off other bipolar medications.
If you don't like Geodon, *mod edit* I'm currently fighting to be taken off antipsychotics and put on more mood stabilisers.
Best of luck to you in finding the right medication.