My solution? Make my own pet slings. Luckily these are relatively easy to make. Start by selecting your fabric. I have multiple slings made of different fabrics. The one in the picture is fleece; I use it in winter. I also have a vinyl sling for wet weather and a light cotton sling for general use. You can pick a fabric that suits your own needs, but it shouldn't have too much stretch to it. A little stretch is fine.
Now you'll have to determine how much fabric you'll need, and this requires a measuring tape. Take a look at the picture and you'll see where the sling needs to fit to support your pet. Around one shoulder and down around the opposite hip. Measure this length and add a couple inches (you'll be stitching the sling together at the shoulder).
So how wide must this sling be? Well, it has to fold in half like a pocket and still have enough room for your pet. The dog in the picture needs a full 18 inches of room, so the fabric had to be at least 36 inches wide. If you're making it for a smaller pet, you'll be able to use a smaller sling. The cat sling I made is actually 24 inches which, which meant I needed a 48-inch wide piece of fabric.
Once you have your fabric, lay it out on a flat surface. Hem the long edges then fold, wrong sides together, to form the pocket. To clarify, the long edges should be touching. Stitch the short edges together so your pocket don't slip during the next step.
To create the sling, you now have to attach the two short ends to each other so the sling will go around your body. Place the sling around you to ensure it's long enough, pin, and stitch. In order to make the sling fit onto your shoulder you'll have to fold the edge accordion-style, but this only adds padding to your shoulder, keeping you more comfortable. Adjust the folds until you're happy. You'll want to add several lines of heavy stitching just in case, then trim the excess (if you have any; I usually don't).
Now you almost have a sling. Well, you have a sling, but if you put it on and put your pet in it, you'll have a pet that slips around inside the sling. There's a lot of empty space in there, after all. So put the sling on, slip your pet inside, and determine where you want him or her to sit. Once you know exactly where and at what angle you want your pet to sit, you can use lines of stitching (decorative or otherwise) to adjust the sling. Stich the pocket closed where you need to and run lines across the sling so your pet don't sneak around to your back (unless you want him to, of course).
Once you're done, you have a pet sling.