You saw, that your mysql disk, using /var/lib/mysql is full. So, you need to know how to free some space and clean up the disk. Usually, you can safely remove the old binlogs.
First of all, you should login to MySQL server and check if there are some binlog files. If you see something like this, they are:
sh-4.2$ ls -la /var/lib/mysql | grep binlog
-rw-rw----. 1 1001230000 1001230000 2959368 Nov 5 04:56 binlog.000001
-rw-rw----. 1 1001230000 1001230000 1074256888 Nov 5 05:04 binlog.000002
-rw-rw----. 1 1001230000 1001230000 623916528 Nov 5 05:12 binlog.000003
-rw-rw----. 1 1001230000 1001230000 16797561 Nov 5 12:57 binlog.000004
-rw-rw----. 1 1001230000 1001230000 385788866 Nov 9 12:38 binlog.000005
-rw-rw----. 1 1001230000 1001230000 484322047 Nov 9 15:16 binlog.000006
-rw-rw----. 1 1001230000 1001230000 424167115 Nov 10 14:04 binlog.000007
-rw-rw----. 1 1001230000 1001230000 58027474 Nov 10 20:00 binlog.000008
-rw-rw----. 1 1001230000 1001230000 13217008 Nov 11 20:00 binlog.000009
-rw-rw----. 1 1001230000 1001230000 179 Nov 11 21:31 binlog.000010
-rw-rw----. 1 1001230000 1001230000 13285404 Nov 12 20:00 binlog.000011
-rw-rw----. 1 1001230000 1001230000 13241773 Nov 13 20:00 binlog.000012
-rw-rw----. 1 1001230000 1001230000 34779 Nov 16 20:00 binlog.000013
-rw-rw----. 1 1001230000 1001230000 359932796 Nov 17 20:00 binlog.000014
-rw-rw----. 1 1001230000 1001230000 22423233 Nov 18 20:00 binlog.000015
-rw-rw----. 1 1001230000 1001230000 86190929 Nov 19 20:00 binlog.000016
-rw-rw----. 1 1001230000 1001230000 39029607 Nov 20 20:00 binlog.000017
-rw-rw----. 1 1001230000 1001230000 101916333 Nov 23 20:00 binlog.000018
-rw-rw----. 1 1001230000 1001230000 32958639 Nov 24 20:00 binlog.000019
-rw-rw----. 1 1001230000 1001230000 40808525 Nov 25 20:00 binlog.000020
-rw-rw----. 1 1001230000 1001230000 40866887 Nov 26 20:00 binlog.000021
-rw-r-----. 1 1001230000 1001230000 156 Dec 1 08:37 binlog.000022
-rw-r-----. 1 1001230000 1001230000 352 Dec 1 08:37 binlog.index
By default, MySQL rotates binlogs and stores them for 30 days. You can reduce this parameter by setting up this parameter at mysqld.cnf:
binlog_expire_logs_seconds = 432000
The value is in seconds, so 432000 means 5 days. You should restart MySQL to apply the settings.
After that you can cleanup old logs. Do not delete files, but login to your MySQL server and execute this query to remove all binlogs:
PURGE BINARY LOGS BEFORE NOW();
All the logs will be purged. As you can see, you can insert your own date here to cleanup not all binlogs, but before a certain date.