I had this exact same problem with my first son, only no one could help me, even the lactation consultant said she'd never heard of it (weird?). I ended up doing my own research to realize I had overactive letdown. My son acted exactly like how you are describing your baby.
What I found was that pumping too much before feeding actually exacerbated the problem, it just signals the body to make yet more milk, which you don't need. What I did was when the milk started to flow, I'd pull the baby off and let the heavy flow go into the nursing pad, once it slowed I put the baby back on.
I would also lie flat on my back while nursing him to slow the flow.
In my case, my body never seemed to adjust and I gave up and just completely pumped all his feedings at 3 months, it was a lot of work but he was happier. The Medela Pump n style is a great pump, doesn't take long. Where you have a toddler at home I don't recommend this, you're busy enough.
Sometimes I'd pump a bottle and give it to him and then later if I became engorged I'd express some of the milk out in the shower. Goodluck, I know it's hard!