In Destination Moon it is at least implied that a slide rule is being used to calculate the trajectory needed to arrive at the Moon. The relevant part of the story is:
“Huh? I wasted reaction mass, so we’re going too fast? That doesn’t make sense." Barnes hooked a foot into the legs of the stool to anchor himself, and did a rough run-through of the problem with slide rule and logarithm table. “Well, boil me in a bucket!" He added humbly, “Doc, I shouldn’t have asked to be skipper. I don’t know enough."
Corley’s worried features softened. “Don’t feel that way, Jim. Nobody knows enough — yet. God knows I’ve put in enough time on theory, but I went ahead and urged you to make the blunder."
“Doc, how important is this? The error is less than one percent. I’d guess that we would reach the Moon about an hour early.”
“And roughly you’d be wrong. Initial speed is critical, Jim; you know that!”
“How critical? When do we reach the Moon?”
Corley looked glumly at the pitiful tools he had with him—a twenty-inch log-log slide rule, seven place tables, a Nautical Almanac, and an office-type calculator which bore the relation to a “giant brain” that a firecracker does to an A-bomb. “I don’t know. I’ll have to put it up to Hastings.” He threw his pencil at the desk top; it bounced off and floated away. “The question is: do we get there at’all?”
In Starman Jones there is no explicit metion of a slide rule being used for astrogation, however there is an implicit reference to it:
There was little to take. Groping in the dark he found the rucksack he used for hunting hikes and stuffed into it his other shirt and his socks. He added Uncle Chet’s circular astrogation slide rule and the piece of volcanic glass his uncle had brought back for him from the Moon. His citizen’s identification card, his toothbrush, and his father’s razor—not that he needed that very often—about completed the plunder.