I thought they were called sanitary engineers ... definitely not professional there! lol...
I'm sure their service contract makes it very clear that you are responsible for having the cans out by 6 a.m. or something and they can just hide behind that, as lousy of customer service as that might be. Options are:
a) If the cans are full of bagged garbage, put on some gloves and nasty clothes and double-bag them into those heavy-duty outdoor bags, load in your wheels, and either dump in a church dumpster (those dumpsters are charged on a flat monthly fee so it won't cost the church anything so long as there' still room for their garbage) or go to the local dump or transfer station (although they'll surely charge you exorbitantly as their pricing usually starts at like half a ton or something).
b) If the cans are full of loose garbage ... I would just bag your garbage for the next week and then put it all out next week. Down here, you get like 3-4 times a year where you can put out "extra" garbage that doesn't fit in the rolly cans but you have to call ahead .. not sure what it's like for you guys.
c) On that note, you may be able to call and schedule an off-cycle pickup ... although, again, that will probably cost exorbitantly.
d) Put Wall on retainer to be your backup garbage-putter-outer-overnight as a safety net going forward
I can see the two full cans though ... my household (and there are just 3 of us ... no dual-production stinky diapers) somehow manages to fill a 96-gallon household waste can, a 96-gallon recyclable waste can, and half of a 64-gallon outdoor waste can every week. It just doesn't seem physically possible to me and yet, there it is...