How to recover posts to new theme from old WP backup

My site was hacked into. It was sending out thousands of emails. I was working with CanvasDreams. I removed the files they identified and install some security plugins. But it happened again and then again. CanvasDreams canceled my account. I went to GoDaddy and decided to start from scratch with a new theme Themetrust’s work theme. I had been using WP widescreen.

I had a lot of posts that I spent a lot of time on on the old site about photography. It would be ashame to lose them. I do have full backup of the old site, it may be from a time where it was infected.
Is there a way for me to recover those old posts? Is it safe to do so?