I have had the same problem with shorts riding up. I like RoadRunner Sports brand compression short. I have tried the "knee" length and the 6 inch length and both are great and don't ride up. They are not super tight compression like other brands of compression shorts though.
Oh my, this is so tough, because I like at least a 4-5 inch inseam and no low rise. I am wearing Nikes--that I have had for years and have a pair of Moving Comfort shorts that I love, BUT they discontinued them. I find this is one of the most frustrating issues for me, more so than running bras and shoes.
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