Function Suddenly Stopped Working

Here is your problem:

"options" => array("Next/Previous Links", "Pagination"), 
"std" => "Next"

Your valid options don’t match your default.

Change this:

"std" => "Next"

…to this:

"std" => "Next/Previous Links"

Then, in your function call, change this:

if( get_option( $shortname . '_next_prev_or_paginate' ) == 'Next' )

…to this:

if( get_option( $shortname . '_next_prev_or_paginate' ) == 'Next/Previous Links' )