Just to follow up... to my surprise, the septic was full. I was
incorrect on the size of the tank, in that it is only a 500 gal tank
(house was built in 1959). What evidently was happening was that
during the night, the tank would drain a bit and everything would be
fine in the morning. Then as the next day arrived and showers were
taken, clothes were washed, etc., the tank would fill to the point of
causing problems.

It's cleaned out now and all is well. However, there was a fair amount
of buildup material that came out of the drain once the tank was
cleaned, so it would be a good idea for me to snake the pipes. Since
it appears that (based on another poster) I can get a full size snake
for the cost relative to once visit, I will look into the full sized
models.

Thanks to all who posted.
Dave