Have you checked AMEX's prequalification page. My EX FICO is 660 and I'm prequal'd for the Delta Gold, Green and Gold charge cards. As far as getting to 700 there can be several factors to improve, utilization, number of accounts, any baddies, inquires. List out what you cards you have, balances and limits and people smarter than me will help you out. Good Luck!
Get you cards paid down to all zero except one or AZEO as we say here. That 40+% is killing you. You want to be less than 8.99%. So 700 is doable. No more apps. AMEX isnt inq or new account friendly. You'll get a boost once you past 2yrs with AAoA's. There ya have it! Good Luck!