I have two google tag manager accounts one mine and another of third party website and I need to put these 2 in this third party website. I tried put two gtm scripts after body
There should be no issues with multiple GTM containers on the same page. Unless you have duplicate codes in the containers which are double firing pixels. You can add those containers after the opening body tag.
To check what is firing on the entire page. Download firebug for firefox and add the "omnibug" extension. The omnibug extension should be in the firefox firebug window when you open it.
When you go the omnibug in the firefox panel, you will see EVERYTHING THAT IS FIRING!
If there are duplicate tags firing in the omnibug box, then you can check the details of the event and figure out which specific tag and delete them from the GTM container that you own.