First, you aren't too old at all. We recently hired a programmer in his late 50s, and man are we glad we did (and btw, age was never a consideration in the first place). He's
great, and we're probably going to use much of his work as a reference point for future apps.
As for getting into it... glad to hear you've had some success with interviews. My biggest bit of advice to you is to get your hands on a good data structures and algorithms book (Java is fine), and learn the crap out of it. Since you like programming, you'll enjoy this, and it'll help you in interviews and as a programmer. In particular, make sure you know:
linked lists
binary trees (be able to code insert, traverse, find on the spot; read about delete but don't worry so much about keeping it all in your head)
sorting (quicksort, mergesort, maybe a few others)
read about balanced trees, b-trees, etc, follow and try out the code, but don't worry about keeping it in your head
Hashmaps (what's a hashing function, how to keep good performance)
Graphs are fun, and once you can code trees, you have all the tech skills, but the algorithms in graphs are very interesting... and now and then does come up in an interview.
SQL - I still get asked about this a lot... know the basic stuff (select, conditions, joins) cold, and also make sure you know some of the grouping functions (HAVING) and how outer joins work. Anything more elaborate will get you points, but probably won't hurt you if you don't know the answer.
Personally, I find it slightly obnoxious that interviewers rely so heavily on this stuff, but it is actually kind of fun. I always review this carefully before an interview. It can make the difference (in fact, I'm pretty sure I missed out on a job because I let it get stale and fumbled around too much... it's a shame, if I'd spent my usual 8 hours preparing, I probably would have rocked it).
It's a lot to learn, but it sounds like you enjoy this anyway.
One other thing... to get experience, try taking on a small software project for your current employer. Once you've done it, you'll have "official" experience. Sounds like I started in a similar spot to you (math major with some CS coursework), so I had to work a little harder to convince employers I could write code. I think EE would probably put you in a pretty good spot, so you really just need to convince them of this thorough those technical screens (which is why I so strongly recommend you read up on data structures).