Next steps in approaching an unresponsive academic
I had a similar experience with a professor while working off an Incomplete.  He did a lot of outside consulting on top of his academic obligations and was one of those people who is simply hard to contact.

In the event that your professor is simply hard to contact I would track down a phone number and try a direct call.  It is possible that your emails got lost in the shuffle or the spam filter.

Alternatively their situation may have changed since you were there: they could be on sabbatical, sick or they may have left the institution.  Do you have any friends still there you could check with? You could also check the course listings to see if they are currently teaching.

Without knowing the professor I think it is unlikely they are trying to duck your request.  If they didn't want to recommend you, there are other ways to handle that such as with a weak recommendation or simply telling you.

I would try to exhaust the other contact options before officially trying to contact them through the department.

Does your new potential school have a portal for recommendation letters?  I know several schools that request the recommendation directly from individuals you specify.