Check out Baleaf on amazon, they meet most of your requirements I think. I have many pairs and really like them. Both the liner and no liner options are good. These cost right around 20-25 so its also a good deal.
I have a few pair of the 5" Baleaf shorts with a zippered pocket on the back. They are some of my favorite shorts and very reasonably priced. They do have a liner but maybe you can check their store to see if they have an unlined version.
I use the baleaf 5" shorts from Amazon. Have the mesh liner and a zipper pocket in the back that I use for trash and maybe a key fob. At $20 a pair, they are by far the best running shorts I've ever run in. link
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BALEAF Men's 5" Running Athletic Shorts Zipper Po… | - | - | 4.5/5.0 |
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BALEAF Men's 5" Running Athletic Shorts Zipper Po… | - | - | 4.5/5.0 |
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BALEAF Men's 5" Running Athletic Shorts Zipper Po… | - | - | 4.5/5.0 |
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Personally I'd opt for proper lined running shorts instead of generic training shorts and underwear. Here's my "go to" cheap shorts and the same thing but a little longer if that's what you prefer.
I like these: https://www.amazon.com/BALEAF-Running-Athletic-Shorts-Zipper/dp/B073P38ZCH
If the $20 is a hard limit, then I have a pair of these that I like almost as much: https://www.amazon.com/Naviskin-Running-Workout-Athletic-Outdoor/dp/B0797RZ4CY
Both dry very quickly.