If it is a date range that you want to check against where basically you
want count if the dates in your database are between 2 dates you can use yet
another criteria field meaning one header for the first criteria you already
have and 2 headers for the date criteria then you could use something like

=">="&MIN(Sheet2!A1:A30)

for the first date criteria and

="<="&MAX(Sheet2!A1:A30)

If it is not between 2 dates but maybe 4 dates in Sheet2 then you can use

=AND(B6="x",ISNUMBER(MATCH(A6,Sheet2!A1:A10,0)))

where B6 is the first cell in the first criteria column and x the criteria
and A6 is the first date in the database and Sheet2!A1:A10 is a range of
dates

Of course you could easily use SUMPRODUCT instead of DCOUNT

=SUMPRODUCT(--(Range1="x"),--(ISNUMBER(MATCH(Range2,Sheet2!A1:A10,0))))

where Range1 is the column in the database that you already got working and
x the criteria, where Range2 is the database range with dates and
Sheet2!A1:A10 the range of dates you want to check against

Both formulas return the same result but the latter is much easier with any
archaic criteria setup