As regarding the Image Magick plugin it runs on my sites with WP 3.9 without any problem. But it needs imagick module installed on the php server.
Under “crop the original image properly before resizing” i mean the following: If you know exactly how wide the image should be in the new layout of your site, then you could crop the originally uploaded image (in photoshop for example) so that it will have a width and height, that is dividable by the number of the targeted image width without any remaining fractionated pixels (applies even to image height). This way the resized and shrinked picture can have the best possible quality. However this method can be really time consuming because you have to do some maths & cropping with all the previously uploaded images.
That is why design and layout really matters and image issues may arise if you change design.
Since last time i examined the page you linked in. The following file is used as featured image: https://mikeshouts.com/wp-content/uploads/2014/04/DeLorme-inReach-Explorer-Satellite-Communicator-672×372.jpg
If you take a look this picture has a lot of jpeg artifacts and fine details are just washed out.
As your content area is wider than 672 pixels, this image will get even scaled up to fit the content area. This leads to further loss on sharpness.
(While the originally uploaded and unmodified picture seems ok for me)
Can you just revert back to your original theme for a few minutes and check exactly which file is used in that case?
My advice is to check if your server has imagick module installed. If so then install the image magick plugin for wordpress and set the compression quality to at least 90% in the plugin settings.
If your server lacks imagick module, then you can only stay on GD-library. In this case you should set compression quality of GD-library at least to 95% in the functions.php of your theme.
In each case you should then regenerate all the picture sizes of the uploaded images with a suitable plugin. Chances are high that you will be satisfied with the image quality after this procedure, but datasize of your images will be larger, taking more time to load your site. That is the trade-off for quality pictures.