We have designed and investigated a new congestion control scheme, called optimal and fully explicit (OFEX) controller. Different from existing relatively explicit controllers, this new scheme can provide not only optimal bandwidth allocation but also a fully explicit congestion signal to sources. It uses the congestion signal from the most congested link instead of the cumulative signal from the flow path. In this way, it overcomes the drawback of relatively explicit controllers exhibiting bias toward multi-bottlenecked users and significantly improves their convergence speed and source throughput performance. Furthermore, our OFEX-controller design considers a dynamic model by proposing a remedial measure against the unpredictable bandwidth changes in contention-based multi-access networks. Compared with former works/controllers, this remedy also effectively reduces the instantaneous queue size in a router and thus significantly improves queuing delay and packet loss performance. We have evaluated the effectiveness of the OFEX controller in OPNET. The experimental comparison with the existing relatively explicit controllers verifies the superiority of our new scheme.