One of the messier (at least for me) aspects of the AppleScript approach is dealing with the time string part of the date-time string that AppleScript stores when you think you are dealing with "just a date." Here's one way to get the the messages sent on a specific date (as opposed to in the last x days before the current date). It also works for multi-day periods (where beginDate and endDate are specified as different dates).
SG
property beginDate : date ("July 6, 2015") --> "Monday, July 6, 2015 at 12:00:00 AM"
property endDate : date ("July 6, 2015") --> "Monday, July 6, 2015 at 12:00:00 AM"
property theSigs : {"Signature1", "Signature2", "Signature3", "Signature4"}
property myMailbox : "Sent Messages"
property myAccount : "myusername@mydomain.com"
set endDate to endDate + (1 * days) --> "Tuesday, July 7, 2015 at 12:00:00 AM"
set pasteStr to ""
repeat with i from 1 to count theSigs
tell application "Mail"
set tgtBox to mailboxmyMailbox of accountmyAccount
set foundMsgs to (tgtBox's messages whose ¬
date sent > beginDate and ¬
date sent < endDate and ¬
content contains theSigs's item i)
--add more filters after the 'whose' between the () above
set pasteStr to pasteStr & ¬
theSigs'sitemi & tab & (countfoundMsgs) & return
end tell
end repeat
set the clipboard topasteStr
--pasteStr -- uncomment to debug