As others have written, contracts which of and concern real estate must be in writing and signed by the person to be enforced against. Google "statute of frauds". Don't start any more threads on the topic, this one is fine.
It is not the contract which is "illegal" in the scenario I envision, it is the attempt to do tax fraud. Why would you make such a complicated legal arraingement verbally?
Why didn't the partner just buy you out and then complete the exchange?
Quit trying to look for a way to enforce something which cannot be enforced. I say again, I think you and your partner were trying to be too clever and get away with something to save on taxes. Now I love saving on taxes and recognize "There are two systems of taxation in our country: one for the informed and one for the uninformed"(Judge Learned Hand) but, this verbal deal is almost certainly an attempt at tax evaision and not tax avoidance. One is legal and the other is not. Going to court will open up the doors of inquiry. Not only will you lose because of the statute of frauds, I believe you will lose because of an attempt at fraud.