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use strict;
use warnings;
...
chomp @seq; # Remove line endings on all entries

printf "%2d: %s\n", length $_, $_ for  sort {length $a <=> length $b} 
+@seq;

Please enter DNA sequence 1:asdf
Please enter DNA sequence 2:zvasdfasf
...
 4: asdf
 7: asdfasg
 9: zvasdfasf