I need a wordpress problem to be fixed

General information for the website: Music webzine

Kind of development: Customization of existing website
Description of requirements/features: We're having an issue with publishing new posts/permalinks. We get a 'page not found' warning every time we try to publish a new post. We are actually stuck right now and are not able to publish any posts. Strangely, for some posts, when we change the titles and play around with new urls, the post is published successfully. But it sometimes takes hours to find a new title and url.

Extra notes:
1) Ours is a new website, and we use a specific wp template. There was no problem for the first few posts. But for over two weeks (about 10 posts), we have this problem everytime we try to publish.

2) What we ask is not only a solution to one specific link, we'd also like to get a permanent solution so we won't have this problem again with new posts.

3) We want to be sure that there will not be any change on previous posts (like social sharing counters reset, or change in permalinks that make them impossible to share etc.)