It appears that your main destination is France/Italy, so you should apply in appropriate French/Italian consulate. If the appointments are all booked, well tough luck.
I, however, would do the following in your case:
- Try contacting other consulates Same country, different consular districts. Maybe they will make a one-time exception.
- Can you apply for a Schengen visa from a country of your citizenship? It may be way faster.
- Some consulates (partially or in whole) outsource visa processing to 3-rd party companies (most notably vfs). Those companies can sneak you in front of the queue sometimes.
- Shift 1-2 days of your journey from Italy to France or vice-versa will allow you to choose a between French and Italian consulates.
- Weigh the financial burden of changing your itinerary to the country with the less visa waiting queue vs cancelling the trip overall. E.g. spend several days in beautiful rural Lithuania. Last time I checked, Lithuanian embassy in the Washington DC accepts walk-ins, and issues their visas promptly.
PS. Appointments for Schengen visas in the US are hard to get for some countries/ some consular districts. The person I know planned her trip half year in advance, and started looking for appointments back then.