To increase the spectral efficiency of wireless networks without requiring full-duplex capability of user devices, a potential solution is the recently proposed three-node full-duplex mode. To realize this potential, networks employing three-node full-duplex transmissions must deal with self-interference and user-to-user interference, which can be managed by frequency channel and power allocation techniques. Whereas previous works investigated either spectral efficient or fair mechanisms, a scheme that balances these two metrics among users is investigated in this paper. This balancing scheme is based on a new solution method of the multi-objective optimization problem to maximize the weighted sum of the per-user spectral efficiency and the minimum spectral efficiency among users. The mixed integer non-linear nature of this problem is dealt by Lagrangian duality. Based on the proposed solution approach, a low-complexity centralized algorithm is developed, which relies on large scale fading measurements that can be advantageously implemented at the base station. Numerical results indicate that the proposed algorithm increases the spectral efficiency and fairness among users without the need of weighting the spectral efficiency. An important conclusion is that managing user-to-user interference by resource assignment and power control is crucial for ensuring spectral efficient and fair operation of full-duplex networks.