How do you align corners of objects to each other?

Mac OS 10.14.6; Adobe Illustrator v. 25.2

I have three curved shapes, which are different colors, that I need to align exactly along their edges and corners. The alignment tools obviously won't work here. I can see gaps between the shapes but when I zoom in to try to precisely line up the edges, the anchor points jump over each other and seem to be repelled by each other. I've tried activating and deactivating Smart Guides, playing with the tolerances, etc., nothing works. Each of the objects is a path with no stroke and a solid fill. I tried to convert the paths to shapes but get the message "Object not converted." See the screenshot below, all three corners should align to one point. 

Also, this may or may not be related: I'm also finding that I have to select objects in the layers panel in order to activate them, that is, I am unable to select objects by clicking on them...sometimes.