What you want is known as a Gray code for integer partitions.
It exists.
See C. D. Savage, Gray code sequences of partitions, Journal of Algorithms 10 (1989) 577-595. Disclaimer: I haven't actually read this paper. Other sources such as this 1997 survey by Savagethis 1997 survey by Savage and these class notesthese class notes don't say what the construction is but say that it's complicated.