How do I know if I’m running a nested shell?

The $SHLVL variable tracks your shell nesting level:

$ echo $SHLVL
1
$ bash
$ echo $SHLVL
2
$ exit
$ echo $SHLVL
1

As an alternative to spawning sub-shells you could push and pop directories from the stack and stay in the same shell:

[root@localhost /old/dir]# pushd /new/dir
/new/dir /old/dir
[root@localhost /new/dir]# popd
/old/dir
[root@localhost /old/dir]#