How scared should I be about DPS?

[Taika]

How stressed should I be about putting out top DPS as a healer?

Not at all. When you're fresh level 70, barely meeting the item level of the content you're doing, and running said content for the first time, no one should expect you to pull "top DPS" numbers. In fact, you are fine to not DPS at all if you feel like you need to focus your efforts on learning the fight, surviving the mechanics, and keeping everyone else alive too. It's a good idea to let others know it's your first time running the content, but even if you don't do that, it's unlikely anyone would say anything at all even if you did exactly 0 DPS throughout the run (unless it means you're not using your Earthly Star at all, maybe ).

First things first: get comfortable with your job and with the fight, then when you know what's coming and when and you are aware of your resources, start adding some DPS. You are likely to find that the more experienced you get with the job, the easier it will also become to start DPSing right away in new content too.

DPSing is a part of playing the healer jobs, but everyone agrees that healing comes first. People don't generally have an issue with healers who are new to content and their jobs and have to focus their efforts on handling the healing part of their job. People do have issues with healers who are experienced with their job and the content they're doing, which means they can get by with as low as 9% healing activity in the content, and they choose to spend the rest of their time doing nothing at all instead of helping their party with DPS when they easily could.

Edit: Talking about DF content here, the normal mode Omega, 24 person raids and so on. When you enter more challenging content with enrages (raid finder content: extremes and savage), you should already be comfortable enough with your job to be able to do some DPS while learning the fight.