Contacting professors via email
Before beginning to email academics at other institutions, I would recommend talking to your tutor/ advisor/ referees first, to find out what route they think is best for you to take. It may well be that they know of certain professors or departments that would be a good fit for you. It would also help your letter writers to understand your goals and hence write a better reference.
Having said that, emailing other academics is a good idea. I have done something similar in the past, with very good results (the advice I received from one professor in particular directly led to me finding a supervisor for my Masters and an offer for a PhD place).
The best thing to do when writing such emails is to keep it short and simple. You are already aware of the time constraints professors have, and an overly long email may be off-putting. Aim to summarise your situation in one or two sentences and emphasise your interest in their field of research (looking carefully at a few of their most recent papers before clicking 'send' is a must).
The resulting email may look something like this:

Dear Professor X,
I am a senior Economics and Maths major at Y University. I have found over the past two years of taking physics and maths courses that my real interest lies in mathematical physics and I would like to do a PhD in this area, possibly at your university. I noticed that you have a recent paper on Z theory, which is something that really interests me. Is it related to the P theorem from this earlier paper of yours?
Due to my non-mathematical background, I am unsure whether I should apply to the maths or physics departments at your university. Can you advise me as to which would be the better fit for my skills and interests?
Many thanks and best wishes,
Junaid Aftab

Be prepared for some people to ignore your email entirely, or to reply with a very short response (e.g. &quot;Thanks for your email, unfortunately I can't help you with this matter at the moment. Best of luck.&quot;). The key is to be persistent. Find someone else to contact. Keep trying.
If you are successful and someone replies with helpful advice, ask if you can follow up with a chat over Skype. I have always found face-to-face conversation more effective than email, and if you are seriously considering doing a PhD under that professor it doesn't hurt to get to know them a little better and get a feel for what they might be like to work with.
Above all, remember that quite a lot of people in senior levels in mathematical physics are interested in mathematical physics and are glad to find interest in mathematical physics in other people.