As you know, a dollar sign marks a variable. You have to take it into account when you are typing it.
You can escape the dollar
./dd.sh "sample\$name.mp4"
or just type it with single quotes
./dd.sh 'sample$name.mp4'
To check if there is a dollar sign in a variable, do
[[ $variable == *\$* ]] && echo 'I HAZ A DOLAR!!!' || echo 'MEH'