If you're seeing a quality degradation across your imported Facebook lives, the problem lies with Facebooks API. The maximum possible resolution is sometimes capped at 360p SD depending on the source of the Facebook Live. We have confirmed this across their API documentation.
Download the HD copy of the Facebook video directly from Facebook and upload into Searchie manually.
Using an external tool (like Streamyard) to stream your Live and use their recording to ensure a higher quality video.