Can Invisalign Realign Upright Lower Molars Due to Extractions?

Molar up-righting is difficult, even for wire braces. If orthodontics is a desire, then I would suggest brackets and wires.
If your bite is a problem, that should be addressed first. It is possible to find a position of your jaw that is comfortable and then use braces to repeat that bite. If you simply move teeth, you might make the bite issue worse.

It may be better to go with conventional braces to upright the molars due to the density of your bone at this point in your life. Consult with an orthodontist that does both and he will give you his opinion.
Invisalign is easier for you to wear and may take longer but conventional braces of few more options for molar uprighting treatment.

Before you get going on braces or anything else, you may want to have a TMJ evaluation. Uprighting a molar is very difficult and best done with braces. Depending on the condition and current state of your mouth, you may benefit from full mouth reconstruction, but I need much more information to help you out with a decision like that.
Also I am curious as to what you mean by painful chewing and preference for soft foods, that is a flag to me that means you might want the TMJ evaluation, and a CT scan to evaluate the joints before you commit to anything. We have CT technology right here in our office and can offer it easily and cost effectively to our patients.

Invisalign can do some uprighting, however it sounds as if you have more major instabilities in your bite than just a minor tilt of the molars. You mentioned in your story that it's your 12-year molars that are missing. The 12-year molars are numbered #2, 25, 28, and 31. I just want to be sure it's your 12 year molars (if so, there should still be a molar in front of the space as well as behind). Then the molars behind the space would be your wisdom teeth and the molars in front of the space would be your 6-year molars (numbered #3, 14, 19 and 30).
If this is the case, I would say the best treatment would be to remove the wisdom teeth and place implants in the 12-year molar spaces. If that is not the case (and it is actually your 6-year molars that are missing), then I would recommend orthodontics (braces with or without mini-implants) first, followed by implants to replace the 6-year molars.

Braces are needed to upright a molar. The biomechanical forces and vectors needed to upright a molar are too much for invisalign. Molars are large teeth with multiple roots and to upright them the roots must be moved through the bone. An alternative to full lower braces may be miniscrews with brackets on the back molar teeth.