They mentioned charging all monthly bills and paying off in full to rack up rewards points on one of your credit cards; wasn't meant as a way to free up money to pay off debts.
You should look into the possibility of getting your phone/internet/cable as a bundle from one service provider to see if you can save a bit each month on those bills.
And like someone else mentioned, $200/mo for cell phone is kinda high unless you have multiple people on a family plan (ie. my AT&T bill has 5 phones on it for ~$250/mo - we split the bill 5 ways mind you 
).
So after adding up all your monthly expenses you should have ~$1000-1200/mo of discretionary income (not including your student loan and car loan payments). If you really focus and don't go out to eat and things like that for awhile you can pay off most of it.
You also said you currently have $800 to pay towards credit cards so if I were you I'd probably pay off the following with that $800. In a way it's also good that your lower balances are on the higher interest cards and your biggest balance is on your lowest interest rate card.
Card Balance Limit Percentage Rage
Amazon Store Card $134 1200 25.99 interest
Belk $75 1200 20% plus unsure
Juniper Mastercard $294 2200 30.24
Paypal Connect Card $193 500 26.99
That should leave you with $104 out of that $800 to put towards either one of these (just put it towards the Walmart one and pay the rest of these two listed below when you can since their interest rates are in the same range). Your next paycheck should leave you with enough to knock out the rest of the balance on those 2.
Chase Freedom $429 500 22.99
Walmart $240 925 20% plus unsure
After that you can start chipping away at these (probably fine to just go from top to bottom since they're all around the same amount; I'd pay off highest to lowest interest myself - Amex > Discover > Chase Amazon > Capital One):
Amex $893 2000 27.24 after one later payment
Chase Amazon $959 1800 21.24
Discover $905 1000 22.9
Capitol One $901 2200 17..90
And then finally pay off the credit union one (highest balance but lowest APR which works in your favor while you pay off the rest first):
Credit Union Visa $4431 5000 9.9
It's gonna take some time and will suck but you gotta stick with it and it'll be a huge weight off your shoulders.
I have ~$40k in student loans hovering over me and I plan to start putting 80-90% of my discretionary income towards it and will be a hermit for the next 2 years or so.
P.S. Sorry for the essay but hope this detailed information and game plan suits you; from a numbers perspective it'll save you the most money in interest charges as well and give you less credit card balances to juggle as you get rid of the smallest balances first.
Can't see it hurting any to transfer the balance from your AMEX to Capital One.
I think Capital One charges an upfront 3% fee when transferring balances but you should double-check. It may be worth it to you to have 1 less bill to juggle. I probably wouldn't myself unless they're offering a 0% APR on balance transfer promo.
That's how it goes, always something popping up to throw a wrench in the gears. Just yesterday I turned my room upside down looking for my brother's car keys and was not a happy camper thinking that I'd have to shell out $300-500 to get the dealership to cut a replacement and program a new key fob. Luckily I found it though.
Good luck and feel free to PM if you have any more questions that come up along the way.