First you really need to figure out what screws up the permalinks, as this can cause a lot more trouble.
I am assuming you have been to Settings > Permalink, and resaved them, as well as eventually try a different format and save.
Beyond that you need to Deactivate all plugins to see if it makes a difference, and if it does reactivate them one by one to find the culprit.
You could most likely edit the proper PHP template, but NEVER edit theme or plugin files, You must do that in a child theme, or all changes will be lost in the next update.
I am not a PHP specialist, but I suspect you need to find the WP filter or function that builds the post title to change it there.
May be @weaver can help to know what file needs to be changed so that Posts displayed with Weaver Show Posts do not have a link in their title.