Bottom line, any time you contact your paint, you have the potential to add swirls. In order to figure out what is causing the swirls, you need to do a full paint correction and remove the current swirls. In other words, you need a clean slate.
PAINT CORRECTION
Start by washing your truck. It is good you are using a two bucket system. Do you have Grit Guards in both buckets? If not, get some! Also, I use two different colored Grit Guards, so I use the same bucket every time for soap and the same bucket gets used for rinsing. I usually throw 2 ounces of All Purpose Cleaner in the wash solution to cut any grease during a strip wash.
Also, I use a larger wash pad (Adam's Merino Wool or Jumbo Pad) on the top half of my car and a smaller pad (Adam's Pro Wash Pad 9x9) on the bottom half. This is important because the larger pads are good for covering more surface, but they are difficult to handle down low and have the potential to pick dirt off the ground when cleaning low vehicles. Also, the bottom half of your vehicle will be the dirtiest part and you do not want to introduce that dirt to the rest of the paint. Wash pads are cheap, relative to your time and paint. Buy two new pads. Buy good ones. honestly, I would skip the drying process and go directly to clay.
Once you're finished washing, clay the truck. Have you clayed the truck yet? Put your hand in a thin plastic bag and feel the paint. Clay the paint and glass until it feels smooth using the bag. With the limited info on this post, I'd be willing to bet the truck hasn't been clayed, or clayed correctly and you are pushing impurities across your paint when washing, drying and waxing. You can use Detail Spray, Car Shampoo, or Rinseless Wash as lubricant. I keep a bucket in my feet as I clay. If you drop the clay and it hits the ground, it's trash. If it falls in the bucket and the bucket has a Grit Guard, you can reuse it.
Once the truck is clayed, use whatever paint correcting system you have. I prefer a machine, but don't know your setup.
Once corrected, a good coat of sealant is required. Additional coats may be a good idea, if this is a daily driver and going to see a lot of use. Sealant will not only prevent impurities from embedding in your paint, but it will also help wash pads glide over the paint. Sealant will also prevent water spots. I apply sealant in a circular motion, either by hand, or machine.
PROCESS IMPROVEMENT
Once your paint is corrected, you need to identify any issues in your cleaning process. There is a 90% chance the damage occurs during the wash and dry process. Hopefully you have two buckets with Grit Guards and two new wash pads at this time.
When I wash a vehicle, I use a straight sweeping motion with the pad, usually going the length of each panel. Use light pressure and let the pad and soap remove the dirt. Using straight sweeps is important, especially if you plan to dry the vehicle with a towel. This will help you determine what is causing the damage.
I do not dry my vehicles with a towel, unless I am in a pinch. I use a Blaster Sidekick or Air Force Master Blaster (I have the Sidekick @ my house, the Master Blaster @ our shop). Anytime you contact your paint, you can scratch it! If you remove contacting your paint from any step, it removes the risk of scratching! That $80 Blaster Sidekick will look cheap compared to your time to correct damage.
If you have to use a towel to dry, I recommend a waffle weave microfiber, like Adam's Great White Drying Towel. When using the towel, spray the paint, one panel at a time, with Detail Spray. I also prime the towel with Detail Spray, to prevent a dry towel from contacting the paint.
When drying, go in straight lines, perpendicular to the direction you went with the wash pad. By doing this, you will be able to determine whether the wash process, the drying process, or both are causing the damage. Sealant should be applied in a circular motion and this will show in the swirls as well.
Hope this helps you figure out your problem!
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