If you drag the dimension you want to show, into the ToolTip Marks card it should appear in the ToolTip.
If I've misunderstood, post back and there are lots of fancy things you can do around tooltips...such as
My first problem was that I had the tooltip box open from the worksheet menu, and couldn't drag-and-drop. Based on your post, I started poking around and dragged the pills onto the tooltip on the marks sheet, but they still didn't show up on the actual tooltip, nor on the drop-down menu in the tooltip editor... until I hit the 'reset' button on the tooltip, which brings in the new fields, but also wipes out any other edits that I had previously made to the tooltip.
Any thoughts on what I'm doing wrong? There has to be a way to add a new dimension to a tooltip without having to revert to default to actually get it to show up
It's nothing you are doing wrong per se! When you create a new viz, items (Dims/Measures) are automatically added to the tool tips as you add them to the Viz (Row/Column Shelf, Colour, Size, Tooltip...etc.), with a default description. If you amend the tool tip, manually, at any time Tableau assumes that you have edited the tooltip to what you want, and stops automatically adding items (even if you drag them to the tool tip shelf). At this point you need to add things automatically by using the 'Insert' options within the tool tip.
What I've found is that, generally, the tool tips are the last thing I do. I create the Viz(es)/Dashboards, and allow the Tootips to build automatically, and then once completely happy I then go back and create the ToolTips manually. It's usually only once I have the Viz(es) finiahed that I know what other info might be useful to the end-user (eg. I might have a dashboard with Sales Value and Volume, and then think that the AUR (Average Unit Retail = Sales Value/Volume) would also be useful). If the user doesn't need the exact values, in my sales/volume example, of sales and volume (i.e. the chart approximate value is good enough) then I might then remove the Sales and Volume from the tool tip, and just leave AUR.
Hope this helps, and makes sense.