My immediate thoughts are that toddler is seriously overtired.
She is waking early into the evening when she would normally be in quite a deep sleep for 3 hours or so.
It also rings bells that she is falling asleep in the car a couple of hours after a long nap!
For a few days can you do a very early bedtime for her?
Can you give her tea and get her bathed before you collect DH. If she falls asleep in the car - put her straight to bed?
I think like many posters, if you can crack her sleep you will feel so much better.
The chances are her poor sleep relates to either chronic sleep deprivation, due to constant night wakings and illness (whiich ime can build up over time) or sometimes (my DS is a candidate) - too long a nap.
In your case initially I would say the former as she is falling asleep a few hours after a nap, and she wakes frequently but briefly in the night.
When my son was/is undertired - he would settle quickly when he went to bed at night, but wake for longish periods and then very early in the morning.
Once she stops waking so frequently you can stop your own napping and will have sufficient time to organise yourself in the day.