Do any of the local running groups have forums? There are a lot of non official runs that most groups do... One of the free groups is more likely for this.
Also, join some online groups. It can be helpful to have a running partner for motivation but not necessary.
To run a 5K (or any distance) without walking, you just need to improve your endurance. That comes from running more. Increase frequency and/or duration of a run. What does your training look like now? Sometimes you need to slow down to do this... but I don't know what you are doing now.