It's not always the sunscreen that's making your skin oily, it could be the combination of it with what you use before it. A lot of people put on too much moisturizer and then apply sunscreen on top, and then you get that sticky effect. I tested a bunch of stuff because I thought it was the sunscreen, but I found out I was using a really heavy moisturizer. Now I use a lighter one, like those gel or serum types, and the difference is huge.
About what type to look for: powder sunscreen, oil-free, or those mattifying ones help a lot if your skin is oily. There are national brands like Sundown, L'Oréal Solar Expertise that make oil-free versions and don't leave that extra shine. But honestly, what works varies a lot from person to person, so it would be kind of risky to recommend one specific one as *the* solution. Maybe it's worth talking to your dermatologist about it, because they know your skin and can suggest something that works with the other products you're using.
Practical tip: test it first with a small amount, like half a coin, wait for it to dry completely before you go out, and see how it goes during the day. Good sunscreen is the one you actually use, so if it feels uncomfortable you're going to stop using it even if it's recommended.