Should I omit weights from my exercise routine and only focus on moderate cardio?

No. Light weights focusing on arms and legs are okay. Avoid weights that are heavy or cause you to strain your core muscles. You can also do exercises that uses the body's own weight as resistance, such as yoga, instead.
No, I would not. The best way to stay in condition is to do a balanced routine of cardiovascular activities and resistance training. The key tends to be in varying the exercises and activities so that you don't get bored. Also, it is ok to focus on different goals depending on what you are trying to achieve. If you have a race coming up, you may want to run more.If you are competing in a sport, than train for that.