I'm hoping for a little advice...

By our own stupidity and lack of self control we have gotten ourself into a credit card mess. And now, because of situations out of our control, we can't pay our way out of it. In April, our oldest daughter (10) has been diagnosed with mental illnesses and our medical bills have shot through the roof. We're having to go to doctors and therapists up to four times a week. Even with our insurance, our copays are killing us. Something has to give. Since we're getting to the point where we can't afford gas and food, I'm thinking it's the cc's that are the most flexible *sigh*
I'm looking at inquiring about settlement from our cc companies. Mostly we've been good about paying them. But we have 30 days late several times on each card. However, each time, we manage to pull it together and pay our minimums. We are maxed out with credit...these cards are all near or totally maxed.
Those minimums are killing us. Again, we're struggling to put food on the table and keep the house electricity on.
We have four major cc:
card 1: Chase (29.9%) $5833 -- $255 monthly minimum
card 2: Chase (6 %) $ 5947 (on a repayment plan) -- 183 monthly min.
card 3: Chase (28%) $4133 -- 128 monthly min.
card 4: Discover (21%) $8279 -- 161 monthly min.
We also have several smaller ones (four with balances of $1,000 and below) but I'm hoping to pay those.
I've done a bunch of reading here and from that I've loosely formulated a plan. I'm thinking that I need to stop paying all four of them and then, after about 60 to 90 days, go to the companies with debt settlement offers.
My questions are these:
1. Should we stop paying on all of them at the same time? Or is it best to stagger the stopping of payments so the 60 to 90 day windows don't land at the same time?
2. Can anyone tell me what I can expect from Chase? I've seen that Discover can be a bear so I'm somewhat prepared for them. I haven't seen anything about Chase though.
Any advice or info anyone can share with me would be terrific. Thanks so much!!