I've had good luck with 4350, Reloader 15 and W760/H414. My standard load for two different .375s is 77 grains of W760 and any bullet between 270 and 300 grains. That usually seems to be a 300 Hornady roundnose for plinking and a 270 TSX for hunting. Reloader 15 gives the higher speeds with 270 grain bullets, 2800 is within reach. I'll likely switch someday, but since the ball powder is so nice for volume loading and still kills things maybe not.
I've noticed a few quirks with the caliber that may be of use to you. First off, they are throated for a 300 grain roundnose, and that can leave a wild amount of free-bore with many other bullets. One of mine will only shoot bullets that reach reasonably close to the lands, and is truely pathetic with anything else. The 300 grain Hornady Roundnose in Interlock and Interbond, the old (and now new again) pattern Hornady solids and encapsulated solids match the throat and shoot well as long as I can keep seating out to the lands. 1700 rounds in 14 months took 120 thou out of the throat if anyone cares.That gets me to the end of the looooong CZ magazine and puts the barrel on borrowed time.The 270 and 300 grain TSXs and 300 grain banded solids do well for hunting loads. My other .375 will shoot everything I tried except the Nosler partition.
Make sure you at least try some standard rifle primers as well as the magnums. It seems to be a bit of a trait that groups can be improved with the milder primers. I can't really think of any reason this should be, except that perhaps it helps keep the burning powder in the case. That sloped shoulder sure doesn't. It has made the difference between "good enough" and 5 under an inch for me.
Don't size your practice cases any more than necessary, many of these chambers are real generous in dimensions.