Digging out a crawlspace into a basement is a job that, if done at all, should be done by a professional. It is NOT a weekend project for a guy with a shovel and a wheelbarrow.
The problem is that you will eventually dig below the foundation of your house. This, obviously, is not good for the long term health of your home (or the short term for that matter).
Even moving the dirt next to your foundation is a dangerous proposition. The dirt inside the crawlspace is exerting a force against the foundation to help offset the force exerted by the outside dirt.
All that helps to hold your foundation in place.
Digging out a crawl will require foundation supports, and an engineering degree to do correctly. It will be a costly procedure if done correctly.
If done incorrectly, it is a disaster.