Thanks for kind offers of help. First let me say that my husband has a Galaxy Tab Pro 10.1 and I have flash working on this tablet for him, so I know how it works to set it up. He watches Amazon Prime videos on this tablet. In my searching I came across the comment that because of the processor of the Tab 3 10.1, streaming video from Amazon will never work (I looked at so many sites and can't remember where it was).
I am using the flash apk provided by moonbutt74.
Here are the different browsers I used and the results - I went into every browser and made sure to set it to desktop mode and enabled flash.
Boat Browser: got a prompt to install Adobe Flash (yes, flash is installed!)
Stock Browser: got a prompt to watch the video on compatible devices, such as Kindle Fire
Firefox Browser: another prompt for compatible devices
Dolphin Browser: compatible device prompt
Puffin Browser came the closest. After selecting the video, the video would load, then the screen would go dark. No video. (This is the paid version of Puffin).
Photon Browser gave the same results as Puffin.
After trying all this, I went into recovery and cleared cache and dalvik. Tried again. Same results. So I am pretty much resigned to no Amazon Prime on this tablet.