Are members of Congress obliged to vote the way their constituents wish, even if they personally disagree?
No. Members of Congress are free to vote however they like, regardless of what their constituents want. It is pretty much inevitable that it would be this way, and it is also, arguably, a good thing that they should do this.
It is inevitable that it should be this way because there is no reliable and cheap way to find out what a representative's constituents think on a given matter. Polls cannot be taken on every issue that comes before Congress.
It is, perhaps, a good thing that it should be this way because representatives (hopefully) are more informed on the issues than their constituents are. Sometimes it may make sense for the representative to vote his or her own conscience because he or she simply knows more.
This is not a danger to democracy because voters can, after all, vote their representative out if he or she goes against their wishes too often.
If you wish to research this further, the idea that representatives should vote their consciences is called the "trustee" model or theory of representation.