Server overload? - Please help!

Guys I really need your help on this one.

I'm running a successful site about advertising on 4.6 with some custom modules, but nothing out of ordinary.

For some reason a month ago I started getting a strange bug. Whenever I go and change any setting in the administration menus the site slows down for minutes and eventually dies. It needs to be restarted. It can be any setting, for example switching on or off a module.

But, this would not happen if I were to change the content of a block or rearrange them. Also, all operations for visitors look and work fine. It is only the admin.

Also, when the site usage at its peak, like Wednesday GMT afternoon it dies on its own without me needing to switch anything in admin. I suspect there can be some kind of operation that puts a burden on the server when many requests are made.

Logically what can be the problem? I have a very similar installation on the very same dedicated server and no such problems occur.

How do I go about such a problem? Is there somebody I can hire who can fix this for me?

Please give me any ideas!
Thank you!
Ivan