Which Oxalis? Some of this family (Yellow Wood Sorrel) is considered a "weed" while other members (False Shamrock) are desirable garden plants.
Generally, Yellow Wood Sorrel (the most common "weedy" member of this family) prefers to grow in poorer soils lacking organic matter, although it will do quite well in soils well amended with OM. Start by getting a good in depth look at the soil in this area and then plant a grass that will tolerate shade once the soil problems are fixed.

The death of the turf grass and take over by oxalis is nature's way of telling you turf grass just isn't likely going to grow there anymore. You could try to find a shade tolerant grass seed and see how it goes. I don't know of any turf grasses that thrive in the shade though, just some that are more tolerant than others.

I think you have to make a choice at this point. Work with nature or against nature. Either is fine, but working against nature will always involve more money, time and labor than working with it.

You might get grass to grow and suppress the oxalis, but only with a shade tolerant variety and if the roots of the tree extend into the area then with increased water and fertilization as well.

A personal recommendation would be replanting the area with shade tolerant wild flowers adaptable to your cold hardiness zone. Get a mix with a lot of varieties because some likely won't be adaptable to the area.

You can pretty much forget about killing the oxalis as it will readily regrow from it's underground parts where it stores energy. Even synthetic herbicides may be of limited use with it. You likely have to outcompete it by choosing plants adaptable to the current conditions that can shade it out by virtue of being taller.