Sorry you're going through this! I've only been in one serious relationship that did not last, so this is just my experience, I know that everyone is different...
When my ex broke up with me after a year of dating, we never fought or anything, it was a friendly breakup -- though that actually made it harder for me, because it was totally unexpected and I thought everything was great with our relationship. We said we would try to remain friends after the breakup, but that turned out to be a huge problem! I was never able to get over the breakup when we continued to spend time together, even though we were usually with groups and not alone. Plus I would even get jealous when he spent time with other girls, even though we were no longer dating. Basically there were just way too many emotions there. This went on for about a year! I think another factor was that we were fairly young -- college sophomores (about 20 years old) at the time of the breakup and it was my first relationship, and I just wasn't very mature at that stage of my life.
Finally I decided the only way to get over him was to pull away from him. I still didn't feel ready to date again, but I thought it was important to just heal myself without diving into another relationship -- looking back I know that if another guy had asked me out while I was still in that healing process, I would have dated him just because I missed having a boyfriend, not necessarily because I liked him. But fortunately no one asked me out and I've never been one to take the initiative
So I started spending more time with girlfriends and less time with the ex. I slowly started to get over him, but what made me get over him completely was going away on a summer internship after my junior year of college. Of course that internship benefited me in several ways and allowed me to mature so much! I had gone to an undergrad college 30 miles from home, and had always gone home on the weekends, and always just hung out with my roommates while I was at college (I was always pretty shy and never had many friends). But on my internship I was all alone, surrounded by lots of other interns who were also alone, and it forced me to quickly open up and make new friends. Within just a week or two, I going out a couple of times a week with new friends (and not always the same group of people, either). It really helped me overcome my shyness, gain confidence in myself knowing that I could still make friends and people did like me for who I am, and of course all the fun I was having made me forget about my ex -- or more accurately, made me put things in perspective and helped me to get over the relationship. (My ex really is a nice guy, but we just weren't compatible and it's obvious now we weren't meant for each other.) When I came back to college at the end of the summer, my roommate instantly noticed a change in my attitude and maturity!
So from my personal experience, I know that dating around would not have helped me get over the relationship -- spending more time in platonic, frienship relationships (and making new friends!) helped me the most!
And incidently, after that summer, I spent the following year of school (my senior year) single and actually enjoying it for the first time in a long time, instead of just longing for a new boyfriend like I used to. My roommate at the time was also single, so we could easily relate in that area and spent a lot of time together. So when I met the man that is now my husband right after graduation and we started dating, I knew that I was dating him because I liked him and not because I just wanted to be in a relationship! We met at the perfect time in my life, when I was actually ready for a serious relationship and confident enough to make decisions regarding our relationship on a much more logical level, rather than just "Oh I can't believe I have a boyfriend! Maybe if I just cling to him he'll never leave me, because no other guy would ever like me!" which is about how I thought when I was dating my ex
Anyways, like I said that is just my experience and it was also very specific to where I was in my life at that time. Everyone's different, and don't worry if it takes time to get over a relationship -- I know it's hard, but if it's not meant to be you will eventually get over it! I would just recommend focusing on other areas of your life that you enjoy, such as friends, family, or work, and if you feel you're ready to date again then you can always keep an eye open for the next guy.
But I don't think that dating just to forget about your ex is best idea -- if nothing else, it's not very fair to the new guy you're dating!