I am currently using file field sources to allow users to select images on our server in the Content Type. However, these are rendering as a download link. Is there something that will let a user either upload or select locally and render as an image in both the full and teaser view?
It sounds like you already have the solution for selecting the image, you just need to control the output of how it is displayed. Are you using the file field instead of the image field? The file field will output by default as a link to the file, rather than as an IMG tag which is what you want.
If you want to keep the field the same but output it as an image, you have two options: you can use an integrated wysiwyg editor or file browser (IMCE is probably the closest to what you're looking for) to assist the user in displaying the file as an image.
Or - you can theme your node template to take the url of the file field resource and output it as an image. This is a little more involved but you can start here with template suggestions and ask more questions if you get stuck.