Blog outreach is the best way to build quality links. However, looking for Authority sites is challenging most of the time. This is why I prefer Postifluence for blog outreach organically. It is giving amazing results till now. It does guest posting on automation and helps me to drive huge organic traffic to my site. You can try it too.

I understand it is so frustrating sometimes to start with. I would say analyze their website pages, pick-up a page that has better performance, do proper keyword analysis, do backlink analysis. If you get any irrelevant and bad backlinks disavow them and check the content whether it is outdated or updated.
CoreSEO has reduced my SEO efforts on Off-page and On-page activities. I am able to analyze backlinks and any other issues hurting my client's ranking and fix them immediately. I must say I have seen noticeable results on the client's website.

1. Fluctuations in domain authority are common, this might be the effect of the bad links got from the directories.
2. Try to get quality links from the local directories.
3. You can always build organic do-follow links through guest posting it helps in building your Domain authority.
   If you're willing to do guest posting, I would say use Postifluence. It will make your guest posting work easier by finding the top ranking guest posting sites and it's quite an affordable tool.

You can always push down the negative content into a Search engine by reverse SEO so it becomes less visible.

After that try creating some genuine backlinks which will not have a negative impact later on. You can try using Postifluence to get high-quality websites for guest posting and build quality dofollow links. I use this tool for guest posting and it's really helping to rank.