This is the exact problem that Meta’s own tools are designed to solve.
The Tool You Need: The all-in-one tool you’re looking for is the Meta Business Suite, which has replaced older tools like Creator Studio for this purpose.
It’s built to let you centrally manage all your content formats, from text-based messages to image posts and video Reels, across both platforms.
The most significant time-saver for you will be the unified inbox, which combines your Facebook Page messages, Instagram DMs, and all the comments on your posts into a single, manageable stream. From the content planner, you can create a single post with its image or video and choose to publish it to both your Facebook and Instagram feeds and Stories simultaneously, or schedule them in advance. Beyond that, it offers combined analytics so you can see how all your content is performing without needing to check two different apps. You can use it on a desktop browser or download the dedicated mobile app to replace the app-switching on your phone.
Cheers,
Jeff