Fortunately there is a pretty simple fix, I had to edit formatting.php in order to keep wpautop() from removing my deliberate breaks. I commented out the following line which replaces two occurrences of BR separated only by whitespace with two newlines:

//$pee = preg_replace('|<br />\s*<br />|', "\n\n", $pee);

Then, I commented out this line found towards the end of the function and replaced it with a version that exempts BRs from being removed:

Commented out:
//$pee = preg_replace('!<br />(\s*</?(?:p|li|div|dl|dd|dt|th|pre|td|ul|ol)[^>]*>)!', '$1', $pee);

Replaced with:
$pee = preg_replace('!(\s*</?(?:p|li|div|dl|dd|dt|th|pre|td|ul|ol)[^>]*>)!', '$1', $pee);

This keeps wordpress from stripping the breaks, but it does not keep Tiny MCE from doing it if you go to visual mode.

update [table_name] set [field_name] =
replace([field_name],'[string_to_find]','[string_to_replace]');

So in this case, to get rid of these weird “” characters, we ran the statement

update wp_posts set post_content = replace(post_content,'ÂÂ','');

Worked perfectly for us, it should for you as well.