I have recently transitioned from a producer to a director and I would like to have more director credits than producer listed in my profile. Can I delete old producer credits, or can I create a new profile as a director only and only include director credits?
No. You should not delete anything that is valid, nor should you create a new page. IMDb is an information source, not a resume service (though, via IMDb Pro, they offer a resume service that you can tailor to your needs). You should not manipulate a page about your works (nor create more than one page per actual person) to look a certain way. It should simply reflect what work for which you've been credited.
The way a page should show you've worked more as a Director is to actually be credited more often for that job. It should list all your valid, eligible credits.
It would confuse users if they look up your page and find two in your name listing different jobs. That implies there are two separate people. All data on your one page should be accurate and reflective of your work, trivia, bio, etc.