I don’t think you need to do more than one away to signal your geographic interest, unless you actually want to check out the programs from the inside (ie to see if this is the kind of environment you’d like to work in).
Psychiatry interviews start as early as mid-October and go on as late as early February at some programs*, but I’d say that peak interviewing time is November-December, so I wouldn’t schedule anything serious during these two months. Also, since applicants often don’t have much control over interview dates, prepare to be flexible.
You sound like a solid applicant. Good luck!
* - some programs actually mention the time period for their interviews on their web sites, as well as days of the week they interview which I found somewhat helpful in my interview planning and scheduling.