So please, link me to an answer if I missed it. But it REALLY seems like the card draws are NOT random. My warrior with 3/4 of his deck as attack cards doesn't nearly draw as many as he should. Same with my cleric. He's got 1/2 a deck worth of healing cards, will go 3-4 hands routinely without drawing one. I must have missed a mechanic on how drawing works. I looked to try to find something about it, but wasn't able too. So please, if anyone has any insight on this, a link or an explanation would be wonderful!
The drawing is random. For reals. One thing that may make it seems like you're drawing fewer attacks or whatever other kind of card you're looking for is that you always draw your default racial move each turn. Those don't come from your deck,.
Like mag said, you get 1 racial movement card every phase (Walk for Dorfs, Run for Humans and Sprint for Elves) and the rest of the cards are drawn from your deck. If you're finding that you're never getting the type of cards you need when you need them, then you should configure your equipment to give you more of those cards, which will increase your chances of drawing them. Sometimes it's a good idea to try to stall the enemy so that you have time to draw the cards that you need. e.g. holding your warrior back until they draw their armour card.
The draws are random in the same way a shuffled deck is random. Your aces may be stacked on top, or on the bottom, or scattered anywhere in between. If all your equipmemt slots are open, you will have a deck of 36 cards. You will draw 3 on your first turn, then 2 on each subsequent turn. This means it would take 18 rounds to draw the first card of your reshuffled deck. Cycling cards (such as traits) will reduce this and make your deck play 'tighter', which is why even negative traits can be important to include. For example, some of my farming wizards carry 14 Firestorms, 4 Ember Bursts, and 11 cycling traits. This means the remaining 7 cards may cause me to go 3 rounds without any wizard drawing a fire burst card. (I know I've gone 2 rounds, but its very rare.) The greater the number of non-cycling, non-attack cards in your deck, the longer you can go without drawing an attack. It's usually a mix, but anything that can happen, will.