Generating All Permutations Without Repetition Using Two Generators

Your question is a special case of the Lovász conjecture, which says that all Cayley graphs are Hamiltonian. According to Igor Pak (Hamiltonian Paths in Cayley Graphs) there is an explicit Hamiltonian Cycle on your two generators which requires only linear space to compute. The construction is purported to reside in this paper (which I cannot access):

R. C. Compton, S. G. Williamson, Doubly adjacent Gray codes for the symmetric group, Linear and Multilinear Algebra 35 (1993), 237–293.

There has been a recent paper by Sawada and Williams where the problem is solved for the harder variant where you always shift in the same direction, i.e., you never require the inverse of generator 2 (Bonus 2 problem). The paper is available here:
https://epubs.siam.org/doi/abs/10.1137/1.9781611975031.37

A demonstration of that algorithm can be run on the Combinatorial Object Server website:
http://page.math.tu-berlin.de/~muetze/cos/
Go to "Permutations", then select "Prefix swaps and rotations (Sawada-Williams)".