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.