The rules with sign and trades is that a team receiving a player in a sign-and-trade deal must be under the tax. The team sending a signed player must take on less salary than what they give out.
In addition, signed and traded players only get the four-year deal they'd get if they signed with the new team directly, which basically means that Melo would have to take a pay cut if he wanted Chicago that badly. If Melo had his heart set on Chicago and New York flipped him for some package centered Deng and the two first rounders they have, I'd understand that. If I don't get a good deal, I'd be okay with letting Melo walk for nothing. Getting "something" is a misnomer. Without first round picks and taking on long-term money only hinders the team. The good news is that if Melo walks, there's a one-year window to absolutely suck and get a good draft pick.
As far as the team structure of the Knicks goes, that was done on purpose with the Bargnani trade. Trying to make significant change with Amare eating a third of the cap is an exercise in futility. There's no reason to bother right now. You can either look at it as the ability to add a better player than Melo later or having Melo go and hitting the reset button when Amar'e and Chandler come off the books.