find matching text and replace next line

One way: Sample file

$ cat file
Cygwin
Unix
Linux
Solaris
AIX

Using sed, replacing the next line after the pattern 'Unix' with 'hi':

$ sed '/Unix/{n;s/.*/hi/}' file
Cygwin
Unix
hi
Solaris
AIX

For your specific question:

$ sed '/<key>ConnectionString<\/key>/{n;s/<string>.*<\/string>/<string>NEW STRING<\/string>/}' your_file
<key>ConnectionString</key>
<string>NEW STRING</string>

This might work for you (GNU sed):

sed '/<key>ConnectionString<\/key>/!b;n;c<string>changed_value</string>' file

!b negates the previous address (regexp) and breaks out of any processing, ending the sed commands, n prints the current line and then reads the next into the pattern space, c changes the current line to the string following the command.

It works. Additionaly is interested to mention that if you write,

sed '/<key>ConnectionString<\/key>/!b;n;n;c<string>changed_value</string>' file

Note the two n's, it replaces after two lines and so forth.