what happens when you whisk sugar with oil?

Oil doesn't really dissolve sugar. The main purpose for the mixing is that in many recipe techniques, especially the muffin method, sugar is treated as a wet ingredient. By premixing the wet ingredients, and premixing the dry ingredients, when the two are combined, you require much less additional mixing to get a homogeneous total batter.

Creaming sugar into butter or shortening helps leavening because the sharp edges of the sugar crystals cut air into the solid fat medium, essentially creating a foam. This then serves as seed air bubbles for the chemical leavening to expand. This effect cannot occur with liquid oils.

Many carrot cakes are pretty dense and oily, although I don't think they should drip when pressed. You might have something else going on in your recipes or technique.

You get a slurry. Depending on the ratio of oil to sugar, you can end up with oil occupying the "voidage" between sugar crystals (think about damp sand) or sugar crystals dispersed in a body or oil. If you have not much oil (determined by crystal size), capillary action would produce a stable "damp" mixture. Again, depending on the type of oil, and type and particle size of the sugar, the suspensions may settle out into two layers over time. Based on a very rough estimate, your 135g in 100ml should yield a very dense suspension, much like a gritty paste.

As pointed out by SAJ14SAJ, sugar is practically insoluble in oil. Your recipes are confusing dispersing with dissolving. You will always end up with a 2-phase 2-component soild-liquid slurry, unlike dissolving (2-component, 1-phase). With butter however, there is always some water in it to dissolve some sugar. After whisking, you end up with a 4-compenent 4-phase (oil, sugar solid, sugar-water solution, air) emulsion (strictly speaking not a foam) or 4-component 3-phase if there is there no excess sugar beyond what is soluble in the water inside the butter. The main difference is that by heating the butter, you will increase the amount of dissolved sugar and vice versa. With just oil and sugar, temperature change would make no difference to the slurry other than the oil viscosity.

I doubt very much if this sugar-oil step is the cause of the dripping oiliness of your cake. The recipes are clearly wrong about dissolving and extending from that, the hope of better mixing using a sugar-oil slurry is flawed. Efficiency of mixing sugar into your dough is about shear force, whether you add oil and sugar separately or pre-mixed would make very little difference, these ingredients still go in as two separate phases. The oil would add some lubricity regardless to aid mixing. If the sugar were actually dissolved in a solvent, one phase, that would make a difference.

I agree that you need to emulsify the eggs and oil like when you make mayonnaise. It works every time for me and no oily cakes or biscuits. First beat the eggs until fluffy. Drip the oil in literally drop for drop until it emulsifies. It will emulsify after about 10 drops. If you skip the drop for drop method your egg and oil will seperate (split). Once the oil and eggs emulsify you can start adding oil faster but still beat in between adding the next lot of oil. Once all the oil is incorporated add the sugar bit by bit until all is used up. Flop proof.