I selected "Default avatar with letters" but on all but one page, the letters only show for a split second and then the default avatar shows. How can I resolve this?

Probably you have some Javascript error on your site, but to check it I need to see your site. Can you please post me your site link?

Also, I would like my users to have the option to either upload their image or use their camera (with crop and drag) for their avatar image. How can I allow both without running the risk of two images showing?

You can set upload method from Components->Easy Profile->Avatar->Options Image->Upload method.

We offer 3 different methods:
- Normal - It is a normal HTML file input
- Crop with Webcam - It allow you to use webcam or upload image (it does not support Drag&Drop)
- Crop with Drag&Drop - It allow you to use Drag&Drop

You can choose only one of these options and there is not way to have 2 images.