There's the concept of having a draft with just a big box of cards that someone already has. You just use that instead of the booster packs and then put all the cards back in the box when the tournament is done.
Indeed. This is known as a "Cube", and doing a draft like this is known as "Cube Drafting" or "Cubing". It is my favorite way to play Magic!
The most common way to make a cube is to fill it with all of the best cards throughout Magic's history. This is what my main cube is. However, you can put together a much cheaper cube with say, only commons and uncommons, or only cards from a certain set, etc.. I also have a "travel cube" that just contains cards I have multiple copies of/aren't super valuable (while still being good cards) to play with at events and things. This lets me relax and not worry about having a mox or dual land stolen or whatever.
I'll be bringing a cube to GDC this year, so maybe you can jump in again. (I think you played my travel cube at TIGJam bobo....)
Also, as far as practicing drafts go, check out
http://draft.bestiaire.org/. It lets you simulate a draft of any set/block against some AI drafters, and will give you a score based on how well it think you did in the draft, based on how highly other people draft the cards you took, etc..
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