I found the issue.
When I initially set up the Cognitive Services I did it on the Free Pricing Tier. This limits the number of sources a project can have to 3. I had since changed it to the Standard Pricing Tier, however, not only does the Cognitive Services have pricing tiers, but the Search Service that it creates also has a pricing tier, which I just found that was still set to the Free Pricing Tier which only allows 3 search indexes to be created. I had 3 existing ones. I removed an old index, created a new project, and added a Q/A pair and it was successful. I also now see this new index in my Search Services list of Index. This is why my other projects were not saving new sources and/or adding new Q/A pairs. I will look into changing the pricing tier for the search services to accommodate my growing list of projects.