How to check if my PC is a zombie and if it is how do I "remove" it?

My PC has been feeling slow and my internet connection seems to lag way more in games and takes longer to load websites, sometimes its really noticeable and sometimes its just fine. I heard it could be because my PC is infected and is used as a "zombie"? Is there any way to check for this and if it happens that it is a zombie, how do I go about removing it? (Preferably without needing to format it or deleting everything).

Extra info about something that happened to me:

I don't know if this is any help but about a week ago I received an email saying my Dropbox account had been accessed from a computer that wasn't mine (Since I haven't used Dropbox in months) This got me kinda worried so I changed my password, but if my PC was indeed hacked or something and they had access to my stuff/passwords, why would someone target my Dropbox (which barely has anything on it) instead of something like Facebook, etc? (This is the only thing that still makes me think that maybe its not infected, otherwise they would have done something worst?)