Bainey - First off, what are you trying to do with this weapon? Do you want a good hunting rifle? Do you want a tactical rifle? What is your end goal.?
First thought, I would suggest looking in the used market. The left handed firearms niche is not small, and while a new factory rifle might be what you want, you might be able to find an overall better value on a weapon that has been built by a left hander.
Outside of that, Savage would be a company that I would recommend you take a look at. They have a couple of good packages right now that are affordable.
Baring that, if you wanted to build a weapon, there are a couple of action makers that make dedicated left hand actions, and also make right hand bolt handled models with a left hand feed & eject system.
There are lots of options available to you, but we need to know what you are trying to achieve before we get too carried away.
By the way, "cross shooting" or trying to use your left eye dominant head with your right hand isn't the best for safety or for inherent accuracy.
I am left handed, but I am right eye dominant, so I learned to shoot right handed very young. This leaves my dominant hand for all other operations, which makes things very nice for me. I don't have the over developed trigger hand problems that some people experience and I can buy "off the rack" for just about anything. I got lucky with genetics.