Why would a cookie recipe start producing puffier, slightly oily cookies over the past few years?
Short answer:
I suggest you try all butter
...and make sure you don't make it too soft before creaming it.

By comparison, here is a famous and popular chocolate chip cookie recipe (just the ingredients, not the process):

2 1/4 cups all-purpose flour
1 teaspoon baking soda
1 teaspoon salt
1 cup (2 sticks) unsalted butter, softened
3/4 cup granulated sugar
3/4 cup packed brown sugar
1 teaspoon vanilla extract
2 large eggs
2 cups chocolate chips

Yours is very similar, and basically is the same except many of the ingredients have been scaled up about 30% (flour, butter/shortening, sugar, eggs).
I would say there are a few ways your results could change. First, if you accidentally buy a different size egg. Your recipe doesn't list the egg size, but it's almost certainly meant for large eggs. As Daniel commented, more egg will lead to a puffier cookie.
Second, if the formulation of the shortening has changed. The oiliness and the bitter aftertaste make me want to blame the shortening. Also, I personally dislike shortening. I suggest you at least try your recipe with all butter, by using 1 and 1/3 cups of fresh, high-quality unsalted butter. I would avoid cultured butter. If you can only find salted butter, you might cut the salt amount in half or omit it. Or you might like the extra saltiness. All-butter cookies come out a bit shorter and denser, but they should still be chewy and crispy and IMHO the flavor can't be beat. Notice that this famous recipe uses all butter. I think shortening actually entered these recipes back in the 1950s when butter might have been more expensive or harder to come by, not because shortening in any way improves the cookie.
Third, there may be something in your process that has changed slightly. The oiliness makes me wonder if the butter and/or shortening is becoming too soft or melted before you cream it with the sugar. It does matter the order in which you add the ingredients and the consistency of the butter and shortening when you cream it with the sugar can matter. Instead of using a microwave, just let the butter sit at room temperature for a little while before creaming it. If you use a stand mixer, you don't even need the butter that soft, just cut it into tablespoon sized pieces and let the mixer beat at it for a minute to soften it up and warm it a little.

Upon further thought:
I'm not sure about all the chemistry that might be going on, but I think Steve has a good thought in his answer when it comes to the flour. I don't agree that you're adding the &quot;wrong&quot; amount of flour, but normal changes in your kitchen environment can have a big change in how flour behaves.
If your problems persist, it could be that your kitchen and/or flour is too damp (Steve mentions temperature, but temperature and relative humidity are closely linked). Normally in cold weather our kitchens are dry, but many things can affect the humidity in a kitchen, such as unusual weather outside or other cooking going on (boiling water can put a lot of moisture into the air). Also, if you've kept your flour for a long time, it may have absorbed moisture and be retaining it even after the kitchen dries out.
Some ways to deal with environmental affects on flour:

Buy fresh flour and don't buy too much of it. Avoid having flour sitting around in your pantry or cupboard for months on end.
Transfer flour as soon as possible after you buy it to an airtight container (this is also a good idea for sugar - both often come in paper bags). Keep the container in a cool, dry place. Not the refrigerator, but a cabinet or cupboard that is not near any heat sources like the stove, oven, or back of the refrigerator.
Measure flour by weight, not volume. Get a good kitchen scale and determine the weight of a cup of your favorite flour. Convert your recipes to weight and then measure using the scale for future baking. If you need to make an adjustment, edit the recipe to show the new weight. If the flour gets too wet or too dry, you will automatically add more or less of it since the weight will change, but not the volume.