How can I add datetime to an existing file name in Linux?

You have created a variable named _now, but later you reference a variable named _nowout. To avoid such issues, use curly braces to delimit variable names:

_now=$(date +"%m_%d_%Y")
_file="${_now}out.log"
touch "$_file.txt"

Note that I have left "$_file.txt" as is, because . is already a variable names delimiter. When in doubt, "${_file}.txt" could be used just as well.

Bonus 1: ${varname} syntax actually provides several useful string operations on variables, in addition to delimiting.

Bonus 2: creative shell escaping and quoting can also be used to delimit variable names. You could quote the variable and the string literal separately (i.e. file="$_now""out.log" or file="$_now"'out.log') or leave one of the parts unquoted (i.e. file=$_now"out.log" or file="$_now"out.log). Finally, you can escape a single character which follows your variable name: file=$_now\out.log. Though I wouldn't recommend reusing these examples without good understanding of shell quoting and escaping rules.

You need to specify the boundaries of the variable name.

_file will be set to $_nowout.log but the shell can not determine if you mean $_now or $_nowout.

Use

_file="${_now}out.log"

to explicitly use the $_now variable.

So the script becomes:

_now=$(date +"%m_%d_%Y")
_file="${_now}out.log"
touch "${_file}.txt"

Which, for the record can be shortened to:

touch "$(date +%m_%d_%Y)out.txt"