Centerfire ammunition price is relative. I have always considered it expensive including the milsurp 308. It is all a matter of perspective.
So you want to shoot? I don't know what you are trying to accomplish with shooting. That kind of helps in understanding what might make sense for you.
In general, I would suggest a couple of things.... (1) Get a good 22 rifle (something like a CZ452/453/455) and shoot that a lot. The skills transfer to your 308 for everything other than recoil tollerance and longer range shooting at the range. But you can get a lot of enjoyment out of shooting a good 22 rifle.
(2) If you want that bigger "bang" consider a 22WMR or 17 HMR and you also get more range. Things like 22 Hornet also cost a lot less per round.
(3) Continue shooting and save your empty brass. If you really like to shoot the 308 win a lot, eventually you will likely get into reloading.