This for an LLC formed in Oregon. If I cannot apply for an EIN being a single member LLC, would I have to use my Social Security Number to represent tax on the LLC, and subsequently assume all business liability?
Yes, you can apply for an EIN, but you don't need one for tax purposes. No, the limited-liability aspect of an LLC does not vanish if you use SSN on your Schedule-C.
It's important to understand that LLC doesn't really mean anything to the IRS. An LLC is taxed as either a sole proprietorship, partnership, C-corporation, or S-corporation based on number of members and how those members elect to be treated for tax purposes. In your case, the reason you don't need an EIN is because the business is just you and isn't taxed as a separate entity, to the IRS you're a sole proprietor whether you had the LLC or not.
The LLC's only real tax benefit is that it gives you flexibility to elect to have your business taxed as a corporation rather than a sole proprietorship (for many, there's no benefit to making that election).
Edit: Answered in the context of taxes, so clarified that you don't need an EIN for tax purposes, they can be useful in some other scenarios.