I would suspect that your theme is built to use the first image attached to the post as a thumbnail image under the title. Your issue isn’t so much a WordPress issue as it is a custom theme issue.
From the post edit screen, when you upload the image, try just saving the image instead of inserting it in to the post. It’ll still be attached to the post ‘under the hood’ so your theme should pick it up, but it won’t display on the post edit screen.
If you go to Media>Library you can see what images are attached to what posts. An image can be attached to a post without displaying in a post.