Your rule actually adds .php to the end.
RewriteRule ^(.*)-new\.php$ $1/new [NC,L]
removes -new.php and changes it to /new
EDIT:
After a while of thinking, if you want users to get redirected, use above and add "R=301" to the brackets as third parameter.
OR, if you actually have links in /page/new schema and files in -new.php schema, then you actually wants to remowe /new and add -new.php. If so, then you asked wrong question. Here you go:
EDIT 2:
You also want to redirect /page to page.php. Now you can do this in two ways:
RewriteRule ^(.*)/new$ $1-new.php [NC,L]
RewriteRule ^(.*)$ $1.php [NC,L]
OR
RewriteRule ^(.*)/new$ $1-new [NC]
RewriteRule ^(.*)$ $1.php [NC,L]
First one are two separate rules, first for /new URLs and then for any other.
Second code is like "first change /new to -new, and then apply next rule"
If you want rule to apply only to urls without dots or anything, then replace (.*) with any other rule like for example ([^\.]*)