Do Shazam and Man of Steel exist in same universe?

Do these two films exist in the same universe under the same franchise?

Yes, and it has been said many times. Such as from indianexpress:

The movie is completely trackable on its own merit. That said, there a lot of fun little nods/homages/Easter eggs that can be enjoyed should one be caught up in the DCEU.”- Zachary Levi

Even director discussed about Aquaman joke too.

From Rrunpee:

According to every reliable internet source, Shazam is indeed set within the larger universe of the DC superheroes, at least from the ‘official DCEU starting point’ with Man of Steel, in 2013.

And they also list other sources too:

Screenrant says: the producers went to great lengths to only show background action figures the DCEU featured to date, holding back on characters they haven’t allotted movie time to yet. It’s a deliberate thing. It’s world-building.

ING.com reports: “So while Superman was battling Zod and then Batman and then Doomsday and then death and then Steppenwolf, Billy Batson was being tossed around the foster care system.”

Also from CBR:

In an interview with Entertainment Weekly, Shazam! producer Peter Safran said "I wouldn’t be surprised to see Shazam play a role in the DCU. He exists in that world." What's more, there is already a plan in place, should the character make further appearances in other films. Since actor Asher Angel, who plays young Billy Batson in the film, will continue to grow up at a rapid pace, there is a contingency to make this all work. "It’s something we’ve considered and have plans in place for sequels that take into account those realities," Safran teased.

I might have missed many other such interviews but I guess you got the point.

Henry Cavill didn't have dates for Shazam, was that the real reason?

Yes, he was not available, from inverse:

“And, of course, I wanted Henry Cavill to pop up. But he was unavailable, which was disappointing.”

Sandberg remembers feeling vexed. “It’s the end of the movie. What are we going to do?” Determined to stick as close as possible to the original idea, he enlisted Zachary Levi’s stunt double to wear Superman’s costume and opted to shoot the Last Son of Krypton only from the neck down. It didn’t calm Sandberg’s nerves, however.

“We had Zack’s stunt double do it, but I was worried. Is this gonna feel cheap? Is this not gonna work? To cut before you see his face, is this gonna feel cheap? Are people gonna feel cheated?”

Later on in post-production, Sandberg finally relaxed when he saw the scene cut together.

“It just made us laugh,” he says. “You see Freddy’s reaction and a hard cut to credits, and it’s just funny. It turned out better than what it was originally, where he sat down and had a little chat.”