A few solutions for you OP:
I sent usually a link to all of those groups 2-3 of my latest blog posts.
Most of the posting was done immedialty after each other, thus no time in between.
1. Don't just share links, write text on the post as well and then put the link at the end of your post (remove the link after the preview). Spin the text a lot.
2. Use clickable images from time to time or just plain images along with text and no links, effectively reducing the number of times you share a link per day on groups.
3. Space out the posting by at least 30 minutes between different group posts, so for example 4 group posts would be completed in 2 hours.
4. Between posts, do other actions as well such as liking/sharing/commenting on your own wall, friends' walls, newsfeed's items, fan pages, messaging your friends, etc... So basically shit a normal person does on FB all day.
5. Use
this plugin to batch rename all image filenames in WP (and also update all references of the images) and then move the entire site to a new domain using a cloning plugin (many around just search for one) or tell your host to do so.
6. Use
this website to batch optimize your images. By doing so, you change the hash code of every image. The hash can also be changed simply by resizing images - search for bulk resizing tools (software or online) on google, there are tons of free tools out there that perform this job. Kraken (the website I linked) has a WP plugin as well, so you can do the batch optimization right within WP without having to download all your images, upload to kraken, optimize and then re-upload again which would take weeks to do.
Warning: The hash code can be so easily changed that you just need to add a single invisible pixel on the image and you will get a new hash. However, I am pretty sure that FB looks way more than the hash in images. Proof of that is the face recognition software that FB currently implements in its platform. I don't know the process that Kraken uses for optimization, but I know it's awesome for the fact that it reduces the size by as much as 60-70% and I use it a lot on SEO. For changing images on the fly, I use MP's extra module to do that and it works so far. But I think you will be OK with kraken as well.