|Hi Sean -
Poison ivy that dies unnaturally (ie weedwhacked) still has urushiol oil in the plant remnants (leaves, stem, etc.). It all depends on environmental conditions as to how long the urushiol remains potent. So yes - it is still potent after being cut but for how long is the question.
Around here we use Brush-B-Gon or RoundUp and spray the plants liberally both above and below the leaf. A little bit of soap like Dawn in the mix will help the herbicide to stay longer on the plant.
As for your dogs, if they get urushiol oil on their coat and you then touch the coat, you'll end up with urushiol oil on you. I've many times ended up with a rash on my upper inner arms as I like to hug my dog and he likes to wander the woods here. In the summer, I tend to have my dog jump in the pool before he gets his hug. Hose your dogs down if you think they may have come in contact with PI.
Hope this helps.