In this paper, we propose two distributed linear convolutional space-time coding (DLC-STC) schemes for full-duplex (FD) asynchronous cooperative communications. The DLC-STC Scheme 1 is for the case of the complete loop channel cancellation, which achieves the full asynchronous cooperative diversity. The DLC-STC Scheme 2 is for the case of the partial loop channel cancellation and amplifying, where some loop signals are used as the self-coding instead of treated as interference to be directly cancelled. We show this scheme can achieve full asynchronous cooperative diversity. We then evaluate the performance of the two schemes when loop channel information is not accurate and present an amplifying factor control method for the DLC-STC Scheme 2 to improve its performance with inaccurate loop channel information. Simulation results show that the DLC-STC Scheme 1 outperforms the DLC-STC Scheme 2 and the delay diversity scheme if perfect or high quality loop channel information is available at the relay, while the DLC-STC Scheme 2 achieves better performance if the loop channel information is imperfect.