If when starting combat an attacker draws 3 combat cards and a defender draws 1, it looks like both the attacker and defender will be drawing cards from their combat deck quite often. Am I right in this?

From reading the rules I go the impression that most combat cards come from your hand and you therefore have more control over the way the skirmishes play out vs. getting lucky with whatever card is pulled from the combat deck.

Yes, and via research order, you can draw additional cards as well to replenish your hand. With good hand management, and varied forces, the impact of luck on combat is rather minimal. One of my favourite things about Starcraft, really, although not all of my friends like it.

Just so. You normally should hold about half or more of your combat deck on your hand. Look into your strategy if not, because there is nothing so bad as not to have the right card for your units.