We present, what is to our knowledge, the first detailed lasing investigation of cryogenic Yb:YLF gain media in the E//a-axis. Compared to the usually employed E//c-axis, the a-axis of Yb:YLF provides a much broader and smooth gain profile, but this comes at the expense of reduced gain product. We have shown that, despite the lower gain, which (i) increases susceptibility to cavity losses, (ii) raises lasing threshold, and (iii) inflates thermal load, efficient and high-power lasing could be achieved in the E//a axis as well. A record continuous-wave (cw) powers above 300 W, cw slope efficiencies of 73%, and a tuning range covering the 995-1020.5 nm region were demonstrated. In quasi-cw lasing experiments, via minimization of thermal effects, slope efficiencies can be scaled up to 85%. In gain-switched operation, sub-50-µs long pulses with a peak power exceeding 2.5 kW at multi-kHz repetition rate were attained. We measured a beam quality factor below 1.5 for laser average powers up to 100 W and below 3 for laser average powers up to 300 W. Power scaling limits due to thermal effects, laser dynamics in pulsed pumping, and multicolor lasing operation potential were also investigated. The detailed results presented in this manuscript will pave the way towards development of high-power and high-energy Yb:YLF oscillators and amplifiers with sub-500-fs pulse duration.