I am not even remotely a theologian, but from what I understand the OO's would not be comfortable with the words "new" and "unique," if that was understood to say that Christ has something different from the nature we have and the nature which God has. I think we are more comfortable with saying He has one "united" nature, which is fully divine and fully human. We believe, like the EO's, that the union did not result in any confusion or change in the natures.
Basically, when the OO and EO theologians got together in the second half of the 20th century, they found that we really do believe the same thing and mean the same thing, only using different terminology. The problem with reunion has to do with the fact that the EO's recognize seven councils and the OO's recognize only three.
One might say, "Why don"t the OO's just accept the additional 4 councils?" or "Why don't the EO's just let go of the last four councils?" However, it is much more complicated than that. The EO's consider all seven councils to be infallible and foundational to their church, and so will not let go of any of them.
The OO's, on the other hand, will likely never accept the other four councils. This is partly because some of these councils condemn persons the OO's consider to be saints and some of the councils use terminology which in the past had been used by heretical groups. (That is basically why the OO's rejected the Council of Chalcedon in the first place and why the division took place.) There are also emotional reasons why the OO's would not want to accept these councils, as they were historically used by the EO's to justify the oppresion, torture and even slaughter of OO's. As recently as a century ago, the czar of Russia was using Chalcedon to justify repressive measures against the Armenians and, at the present time, the Georgians are using it to close down or confiscate Armenian churches.
If you want to know the history of all this and how the division came about, a good source is The Council of Chalcedon Re-Examined, by V.C. Samuel. You can get it through amazon.com.
My advice however, is not to get hung up over this issue. If you are exploring Orthodoxy with the idea of perhaps one day converting, this is a red herring which will just get you off track. Just find an Orthodox church near you and start going to services and talking to the priest. You'll learn more that way than getting involved in these aggravating internet debates.