I have Java-based web server, and I also have DAO singleton object with method, whose SQL operations\' logic must be synchronized in some way in order to guarantee data integri
Yes, using transactions is better. With synchronizing in your code, locking on the class, the scope of that lock is your classloader, and standing up a second instance of your application will invalidate your locking, because the two instances are using different locks.
With database transactions you can have multiple instances of your application and the database treats all the transactions the same.
Also with databases you have options like dialing down the isolation level to no higher than what you need for that transaction, or using row-level locking. Those are harder to implement in code and you're still stuck with not being able to deploy a second instance.