What can I do to finish intersecting plane models with minimal buckling?
Yes, paper may play a role here. Typical Kami paper (I assume this is what you mean by "square origami paper stock") is rather thin which means it is easy for the outer, visible layer to become crumpled and embossed with the pattern of the layers below when you squeeze multiple layers together. An example is visible as a diagonal line on the yellow unit in the picture. Thin paper may also wrinkle and deform as you force the flaps inside pockets. In my opinion even plain colored copy paper is often better since it's somewhat thicker and stiffer. You could also try stiffer papers such as Tant or Elephant Hide. They will make it harder to leave marks on the paper as you move it around, but on the other hand the extra stiffness may make connecting the units even harder as it's harder to bend the units as you wrestle with them.

The other important aspect is technique. With tightly-knit models it is usually really hard to insert the last unit or two, so some level of imperfection is probably unavoidable. But there are some things you can do to reduce visible artifacts:


Strive for more symmetry as you fold. Instead of just inserting the last flap into its pocket when everything else is already in place, try to close the last vertex where three modules meet in a single step. So instead of two tabs fully inserted and one not at all, try to start with a loose assembly where all three flaps are just barely inserted into their pockets. Then, slowly try to tighten it, inserting the flaps deeper and deeper, all at the same time and trying to keep is as symmetric as possible. At one point the flaps should snap and all fall into their positions simultaneously. This technique requires at least moderately stiff paper and might not work with Kami (but give it a try). It may require a lot of coordination but often this technique allows one to close corners which would be impossible to close by manipulating just a single flap.
When folding, sometimes you have to make inexact folds on purpose in order to accommodate paper thickness. Diagrams are sometimes drawn with "perfect", zero-thickness paper in mind. Actual paper has non-negligible thickness and when you fold five layers, they are about ½ mm thick which can already be significant. So sometimes when you fold an edge to a point, you should actually fold a little bit before it so that it gets into the right place later when more paper layers accumulate.
Practice. Some tricks are just muscle memory and hard to explain to others but you will come up with such tricks yourself and develop techniques that suit you well as you fold more.
Cheat. Sometimes you can leave out the last flap by folding it back below the unit instead of into its pocket, and no one will notice. In a model with strong locks, one missing lock will not significantly affect the model's stability and may not be easy to spot visually, either.