You’re correct, when you update the permalinks, the htaccess file also updates.
When you create or update a “pretty” permalink structure, WordPress
will generate rewrite rules and attempt to insert them into the proper
.htaccess file. If it can’t, it will say something like “You should
update your .htaccess now” and print out the rules for you to copy and
paste into the file (put them at the end).
Source: WordPress Codex.
First, verify that the htaccess file has the appropriate lines. Then verify that your hosting server has all the requirements (apache, mod_rewrite etc). Finally, disable all plugins and see if it works like that. If it works, enable one plugin at a time to find the one that’s causing the issue.