For a higher limit, I suspect your best bet is a credit union but you'd have to navigate the membership process first. And, in some cases (e.g. NFCU) debt to income ratio is very important.
All of that said, my three recommendaitons to research are as follows:
I just made a big purchase spread out on several cards. My utilization was 10% before the recent big purchase. The new high balances have not hit my credit report yet.
I can pay it all off within five months with out any refinancing it. I was just hoping to get a balance transfer card.
The low utilization on my credit report is at the moment before the big purchase is reported to the credit report.