I found an addon module that improves posting images.
If you're sharp eyed you'll see the phrase "Insert image/file" below the text area when making a posting. Click on that. It's almost straightforward.
First you click the Browse button. That lets you browse your disk for a file to upload. Once you find the file to upload, click OK in the file dialog, and then click Upload button. It'll take a couple moments to upload.
The next step is that the image will be listed towards the top of the window. Click on the add button in the line listing the image. Voila, the HTML for inserting the image will be included in the textarea.