This paper studies a two-user cooperative non-orthogonal multiple access network with a novel cooperation scheme, where the source communicates with the near user directly or through the help of $K$ relays, while the communication to the far user relies on the help of the near user or the $K$ relays. For cases with decode-and-forward (DF) relays and amplify-and-forward (AF) relays, we propose two different two-phase transmission protocols. In each protocol, the near user sends a two-bit feedback (in DF-relay case) or a one-bit feedback (in AF-relay case) to inform relays of its detection results in the first phase. Then based on the limited feedback and the available channel state information at each relay, a relay selection strategy with adaptive selection criteria is proposed for each transmission protocol to minimize the system outage probability (SOP). Tight-approximated as well as asymptotic closed-form expressions of the SOP for both transmission protocols are derived. Asymptotic results illustrate that our proposed adaptive relay selection strategies have the full diversity order of $K+1$. Furthermore, we analytically prove that when DF relays are used, SOP is smaller than that when AF relays are used.