htaccess - change file dir with filter - .htaccess

Question

I'm currently trying to make a htaccess thing to find the file in new/ if the request uri not contains api or list. I tried to do a thing like this but, it shows 403 error. Are there a easier way to filter, or?
RewriteEngine on
RewriteCond %{REQUEST_URI} /api/ [NC]
RewriteCond %{REQUEST_URI} /list/ [NC]
RewriteRule ^/(.*)$/ new/$1 [L, R]
Full htaccess code
#remove html file extension
RewriteEngine on
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.html -f
RewriteRule ^(.*)$ $1.html [NC,L]
#remove php file extension
RewriteEngine on
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.php -f
RewriteRule ^(.*)$ $1.php [NC,L]
#set max post size
php_value memory_limit 2048M
php_value max_execution_time 3000
php_value max_input_time 6000
php_value post_max_size 800M
php_value upload_max_filesize 200M
#Caching schema
<FilesMatch "\.(png|svg|css|eot|ttf|woff|woff2|otf|js|css)$">
Header set Cache-Control "public, max-age=86400000"
</FilesMatch>
DirectoryIndex index.html
#Custom 404 errors
ErrorDocument 404 /404.html
#Prevent viewing of .htaccess file
<Files .htaccess>
order allow,deny
deny from all
</Files>
<Files error_log>
order allow,deny
deny from all
</Files>
#Prevent directory listings
Options All -Indexes

Answer 1

EDIT: With OP's full htaccess file please try following.
#remove html file extension
RewriteEngine on
##For external redirect rule.
RewriteCond %{THE_REQUEST} !\s/(?:api|list).*\s [NC]
RewriteRule ^(.*)/?$ new/$1 [L,R=301]
##For internal redirect rule.
RewriteRule ^new/(.*)/?$ $1 [L,NC]
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.html -f
RewriteRule ^(.*)/?$ $1.html [NC,L]
#remove php file extension
RewriteEngine on
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.php -f
RewriteRule ^(.*)/?$ $1.php [QSA,NC,L]
#set max post size
php_value memory_limit 2048M
php_value max_execution_time 3000
php_value max_input_time 6000
php_value post_max_size 800M
php_value upload_max_filesize 200M
#Caching schema
<FilesMatch "\.(png|svg|css|eot|ttf|woff|woff2|otf|js|css)$">
Header set Cache-Control "public, max-age=86400000"
</FilesMatch>
DirectoryIndex index.html
#Custom 404 errors
ErrorDocument 404 /404.html
#Prevent viewing of .htaccess file
<Files .htaccess>
order allow,deny
deny from all
</Files>
<Files error_log>
order allow,deny
deny from all
</Files>
#Prevent directory listings
Options All -Indexes
With your shown attempts/rules, could you please try following once. Please make sure to clear browser cache before testing your URLs.
RewriteEngine on
##For external redirect rule.
RewriteCond %{REQUEST_URI} !^/(?:api|list)/? [NC]
RewriteRule ^(.*)/?$ new/$1 [L,R=301]
##For internal redirect rule.
RewriteRule ^new/(.*)/?$ $1 [L,NC]
OR try following rules set: Please make sure try only 1 set at a time, either above OR following.
RewriteEngine on
##For external redirect rule.
RewriteCond %{THE_REQUEST} !\s/(?:api|list).*\s [NC]
RewriteRule ^(.*)/?$ new/$1 [L,R=301]
##For internal redirect rule.
RewriteRule ^new/(.*)/?$ $1 [L,NC]

Answer 2

You may try these rules in your site root .htaccess:
RewriteEngine On
# redirect to /new if not /api, /list, /new and file exists in /new
RewriteCond %{THE_REQUEST} !\s/+(new|api|list)/ [NC]
RewriteCond %{DOCUMENT_ROOT}/new/$0 -f
RewriteRule .* /new/$0 [L,R=301,NE]
## remaining rules go below this line ##
# remove html file extension
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}.html -f
RewriteRule ^(.*?)/?$ $1.html [L]
# remove php file extension
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}.php -f
RewriteRule ^(.*?)/?$ $1.php [L]
Make sure to test it after clearing your browser cache or test from a new browser.