In this paper, we demonstrate that despite the power constraints on all transmit nodes caused by the interference temperature limit (ITL) imposed by the primary user, power domain cooperative non-orthogonal multiple access (NOMA) can ensure much higher throughput in underlay cognitive radio networks than conventional orthogonal multiple access (OMA), provided the network is adaptive. We use intelligent user-selection and switching between cooperative NOMA, (non-cooperative) NOMA and OMA to ensure high throughput at the selected far-user while ensuring a desired performance at the selected near-user. The suggested user and mode selection scheme is simple and easy to implement, and yet ensures performance comparable to that attained by a scheme using more information. When the ITL is broadcast by the primary receiver in every coherence interval based on the primary channel, we show that large gains in throughput accrue. Accurate expressions are derived for throughput of the near and far users with these schemes using peak power and ITL-based power constraints. Performance with imperfect successive interference cancellation is also analyzed. Useful asymptotic expressions are derived. Computer simulation results validate the derived expressions.