Often, the production of hair oil adjusts naturally over a few months to a change in hair routine. So, possibly, your hair will adjust by itself.
In the meantime, I suggest using the minimum water and doing the minimum necessary amount of washing for wudu (rather than anything extra/mustahhab).
This will vary between schools of thought. For instance, the 4 Sunni schools of thought require wiping the entire head (but have some minor differences regarding wudu between each other). The Twelver Shi'i school of thought requires wiping a quarter of the head with 1-3 fingers by using the remaining water from wudu (not taking extra water).
Historically, most people did not wash their hair daily, and certainly did not have access to commercially produced shampoos, so, in and of itself, performing wudu regularly does not usually require people to wash their hair more frequently (although individual circumstances vary).
This is venturing outside the scope of responses offered here, but you could look into advice online by people starting the "no shampoo (no 'poo)" routine in reducing oiliness. Possibly dry shampoo might be helpful. Of course you have to decide what is best for yourself, your health, and your circumstances.
Lastly... doing wudu doesn't usually make people's hair oily. It is possible that this is happening for yourself, since everyone is different. However, if you haven't already, you could look into other causes, such as hormones, stress, diet, or an underlying health condition. If you have recently converted to Islam, this could be an underlying stressor, if it has come with shifts in your life or challenges.
Best wishes!
