No need to use grep. ps in Android can filter by COMM value (last 15 characters of the package name in case of java app)
Let’s say we want to check if com.android.phone is running:
adb shell ps m.android.phone
USER PID PPID VSIZE RSS WCHAN PC NAME
radio 1389 277 515960 33964 ffffffff 4024c270 S com.android.phone
Filtering by COMM value option has been removed from ps in Android 7.0. To check for a running process by name in Android 7.0 you can use pidof command:
adb shell pidof com.android.phone
It returns the PID if such process was found or an empty string otherwise.