If you plan to use this rifle in Africa for large or dangerous game (lion, buffalo, etc.) the .375 is clearly the better choice over a .338. For African plains game and anything in North America the .338 is probably a better choice. If you don't plan on large African game in the next few years the .338 will serve you very well.

I have more than one .375 and more than one .338 and have hunted with them all extensively. I use both a .375 H&H and a .375 Ruger and am very happy with both cartridges. In your budget you can find an excellent Ruger Hawkeye (I have seen new ones for around $800 and very lightly used ones for as low as $600), and for a few hundred dollars more an excellent FN made Winchester Safari Express (I think I paid about $1,200 for mine). If I was going to start all over again I think I would just buy the Hawkeye and use the rest of the cash to go hunting with it. The Ruger .375 is probably a better designed cartridge and ammo is no longer a problem to find, and the H&H has seniority and nostalgia.

All of my bolt rifles that are .375 or larger are CRF. For anything you would shoot with a .338 either PF or CRF would do the job equally well.