Don't forget that with an expiring contract/amnesty clause/whatever, you DO NOT get to use whatever money is expiring for free agents. You have to be BELOW the cap by whatever amount you want to use.
For example, the C's are at, what, 70 million dollars, and the cap is about 55 or so. Even if the C's had 15 million in expiring deals, they could let those expire, and be down to 55 million dollars, and have exactly Zero dollars to spend on free agents.
Now look at your scenario: Say the Cs are at 70 million in contracts, which does not count Davis as he is a FA. So they sign davis (bringing contracts to, say, 75 million), and trade him for a guy making a little more (say six million)...then they amnesty him. Well, that new guy's contract now doesn't count against the cap, so with his 6 million and the previous 15 million expiring, the C's now have 21 MILLION in expiring deals! But, remember they were already over the cap, and had to go even further over the cap to Sign and Trade Davis, so all their expiring deals expire and they get down to...55 million, or right at the cap.
Summary/Conclusion: For the purposes of Free Agent Signings, the amount of expiring contracts is IRRELEVANT; signing free agents is all about how far below the cap you can get.