Edit: Based on the comments, I will add a bit of clarification to our situation.
For the coastal area we live in, $84K is a bit over the median income (which is not adjusted for number of children).
Our credit card debt is a recent situation - while we have used credit cards to build credit since we married, we used to pay them off each month until the birth of our third baby. Medical bills, the need to buy a new car (which was secondhand and a great price, but still), lowered income during maternity leave, and a longer gap between graduate school stipend and postdoc salary than we had anticipated combined to get rid of our savings and put us in debt.
Consumer goods are not a huge part of the problem. Currently, healthcare premiums, taxes, rent, utilities, childcare, auto insurance, commute costs, and gas, all of which are relatively fixed, make up 84% of our monthly income (this isn't including credit card payments or groceries). Our grocery bills have gone up recently due to a child with special dietary needs. We rarely buy takeout or eat in restaurants and rarely buy clothes or other consumer goods.
Until recently, I was calculating our monthly income by dividing our yearly income by 12 - although this meant that our budget was technically covered, I still found myself using credit cards occasionally. I realized that this is due to the fact that most months only have two Fridays for paychecks so our monthly income is several hundred dollars less than I had been calculating - taking us from result:happiness to result:misery.
There is no realistic way to downsize on car/home - they are all relatively inexpensive and as our cars are already quite old and our house is quite small they can't be downsized further without considerable loss in quality of life.
Regarding asking family for help - they are not so well off that a $13,000 loan would be easy for them - when I wrote that we could ask them for help I meant that in the case of an emergency (medical emergency, job loss, etc.) we can probably count on them which is why my husband feels that a full emergency fund (3-6 months monthly income) is not a necessity for us.
I am a big fan of David Ramsey - however, it is difficult to pursue paying off debt with gazelle-like intensity when one's spouse disagrees with the premise that debt is bad.
Our Plan
Thank you to all of you for your advice. We had a long discussion about how to handle our finances and were able to come to an agreement on how to view debt in general and on how to manage our finances in the short term:
Debt consolidation - either by calling the credit card companies and asking for a better deal or getting a new card with low interest on balance transfers (and freezing all the cards in ice to prevent using them as much as possible), or possibly a personal loan if we qualify.
Try to reduce somewhat negotiable fixed expenses (utilities, auto insurance, etc.).
Reduce (already-small) spending on consumer goods as much as we can.
Try to maximize income relatively easily (e.g. asking for a raise, tutoring nearby children during the evening, etc.) without sacrificing on personal/family time.