I want to go Auburn... help please

My hair's natural color is med brown. It's fine. But very healthy.

I have it dyed a med/red brown at the moment. But I would love to have more red in it.   I have used drug store brand dyes. And dyes from sallys beauty.  I really don't want to use bleach on my hair.  I don't want to damage it and dry it out.

Is there a way I could get a nice med/dark Auburn without bleaching?

 

Also.... I was thinking about just buying a drug store dark auburn color. But what I was afraid of was, my hair color going TOO DARK.  OR even black... I didn't know if that would happen, since I'm putting it over my darkish colored hair already.

 

Any advice would help me out.

Thanks so much! =)