If you want to collect all date values, you will need an array, not a scalar:

my @dates;
foreach $odate (@logs) {
    if ($odate =~ /\bdate=(\d{1,4}-\d{1,2}-\d{1,2})/i) {
        push @dates, $1;
        print "Found $1 as a new date\n";
    }
}

print "I collected the following dates: @dates\n";
for my $date (@dates) {
    print "Processing $date\n";
};
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Also, while the following code works, it could be written differently:

while (@logs = <FILE>) {
    foreach (@logs) {
...
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The above code reads the contents of FILE into @logs in one go, and then iterates over that list one by one. You can roll the two "loops" into one by eliminating @logs and using:

while (my $odate = <FILE>) {
   ...
};
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