Thank you for the explanation, things are much clearer now.
I'm sorry that your insurance won't cover iv sedation, but you should also know that it is possible to get through several extractions while being conscious too and they are much quicker than any other procedures. I also assume that your dentist is a competent one and if the procedure would be too complicated, he/she would at least schedule you for two separate appointments. If he/she didn't, then it will be because it will be easy.
First of all: you shouldn't feel any pain during the extractions and it's also possible for the shots to be painfree, so do not worry about this one. Tell your dentist about your worries beforehand and let him/her explain to you what she/he will do to make it as comfortable as possible.
For your anxiety you have nitrous. It should relax you. The important thing is to breathe through your nose and to give your dental team feedback about if you feel it's working or not so that they can make adjustments.
I would like to encourage you to tell your dental team what exactly you are scared of and do not hesitate to let them know if you feel any discomfort during the procedure. Agree to a stop signal to get a breather whenever the anxiety gets too much or you feel anything. They need your feedback to make it comfortable for you and the more information about your worries they have the better they can put you at ease.
Keeping my fingers crossed for you, all the best for the procedure may it all go smoothly, quickly and comfortably.

Take care and keep us posted.