While Instagram is beginning to roll out a way to post from a computer directly, without any other apps or manipulating a browser, that has not yet been fully implemented. My method is to put images into a square "canvas", which can be done in Photoshop, but is even easier to do in Lightroom.
Instagram suggests a basic dimension of 1080 x 1080 pixels. Using the print module, I make a "page" with those dimensions containing one photo each, and export it as a file as a JPG. Works like a charm! I typically post 10 images as a group (the maximum Instagram allows), and this process works fine for that, as well. No matter the shape of the image, you always see the entire thing.
After that, I do a similar process to yourself, which is to use Air Drop (or email) to send the images to my phone, then I pick them up from them there using the Instagram app. To make it even easier for myself, I write all my captions, text, and hashtags in the Notes app on my computer, which also appears on my phone. I simply copy-and-past all the text and drop it into the post on the phone.
I am really looking forward to the day when I can post directly from the computer, but in the meantime, once you get used to it, this process only takes a few minutes.