To extract the <name> part of the file name, provided the dot separates <name> from <date> you can use parameter substitution in this way

gistfile1.sh

#!/bin/bash
cd /path/to/backup/directory
for file in *.tgz
do
  name=${file%%[.]*}
  echo $name
  if [ $name = $current_backup_name ]
  then
    # do new backup here
    # delete older one here
  fi
done
where current_backup_name will be the name of the backup you are currently processing, against which you want to find an existing backup. The ${file%%[.]*} substitution, will strip the longest part of the variable "file" from the back end, matching the pattern .* (dot followed by any number of other characters).