Oneworld doesn't provide a public contact, and even if they did, this is nothing to do with them. I am afraid you are talking with someone at Qatar who is either completely clueless or being deliberately obstructive.
If you have flown any part of the ticket, then control of the ticket can pass to the next airline on the ticket. Some airlines (AA, BA, CX, QF at least) will be willing to take over the ticket and re-issue it for you on their stock. [However, if they do they may do a tax/fee/surcharge recalculation on the existing segments (even before changes are made). BA is quite famous for collecting fuel surcharges that other validating carriers omit. Therefore you may wish to keep trying with Qatar.] Otherwise, Qatar retains control of the ticket.
Unlike most tickets, the oneworld Explorer fares cannot be issued automatically and must be manually priced and issued by hand. This is a rare skill that is only possessed by a few people nowadays. Therefore the front line agents are powerless to help you. You need to talk to a ticket desk. How did you book this in the first place? Do you have a contact for them? I suggest you start with that contact.
Your eticket receipt should include the issuing office and their contact information. If you did not get an eticket receipt, you may use a tool like the MyFlights app to find the issuing office information in the PNR.
There is no penalty for changing the dates within the ticket validity period, but a local service fee may apply. However if you are making changes before the departure of the first flight and the fare has increased since you bought the ticket, the difference will be payable.