Im anemic, but didn't know till about 3-4 after I got my first period. I've always struggled with blow outs and ruining pants. Here are a few suggestions you can try to stop leaks:

1. Get the longer pads! 

2. Use over night pads.

3. Use tampons. I never used them till last year and they're so fantastic, I really wish I would've used them when I first got my period.

4. Make sure you're not anemic! Once I found out I was anemic and started taking iron pills my period got SO much better! 

5.  If youre having blow outs double up the pads, kinda like a diaper. It sucks, but if you dont like tampons it's your only choice besides the cup.

Seconding Oz's recommendation:  Menstrual cups are great!  Give yourself some time to get used to them, and if you feel you need backup for one, I strongly recommend cloth pads.  For one thing, I'm confident you won't need them with the cup, but they're nice to have, and you won't produce any waste.  Sometimes, when my period is really light, pads are nicer to use.

Just like the myriad of choices you have for cups, you have your options with pads.  To name a few:

GladRags
Lunapads (available on this site!)
MamaCloth
NewMoon Pads
Sckoon 

Aaand…that's all I know about cloth pads, right now.  I have Glad, Luna, and Sckoon, myself.

I haven't had any serious leek problem ever since I started using menstrual cups, so I strongly recommend you use them. They can take much heavier flow than most pads, specially if you combine them with pads, and the best thing is that when they begin to overflow unexpectedly, you can usually feel it, because since you can keep yourself dry and clean most of the time you develop a greater sensibility to the flow. Nowadays I can always feel it when I'm starting to leek, and there's usually plenty of time to find a bathroom before the flow even get to panties/pads. So my advice is: if you haven't already, get yourself a menstrual cup.