However, if I want to read another text file line by line in the same perl script, redefining $/ = "\n" doesn't work.

It will depend on whether you are reading the other file in the same lexical scope. If you are not then you can localise the scope of your redefine of $/; you should be doing this anyway as a habit to avoid the side effects you describe.

{
    local $/ = q{}; # paragraph mode in this scope
    while ( <$file1FH> )
    {
        # do something with the multi-line record
        # from file1
        ...
    }
}
...
# $/ now back to normal
while ( <$file2FH> )
{
    # do something with a line from file2
    ...
}
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I hope this is helpful.