Teresa, I had the same complaint when making pasta. Not always but from time to time. Depends on a lot of things.
First, from what I experienced, the choise of flour is very important. Italians use "typo 00" aka "doppio zero" which is sometimes hard to find outside Italy.
A good alternative is the "strong" flour used in bakeries for baking bread. It contains a high amount of gluten, just like doppio zero. Someone advised me to use what we call over here "fluid flour", which is a very finely milled semolina. Don't know wether this stuff is available in the US or elsewhere. It flows easily through your fingers when you lift some. BTW, Italians use "farina de semola di grano duro". Semola, or semolina, means semi + molinare; meaning half + milled (-ish) kind of flour. Flows also easily through your fingers, unlike softer flour. Grano duro means hard grains (which make "strong" flour ,I guess??).
Second, there's the processing of the ingredients. The simplest standard ratio to never forget is 100 gram flour + 1 BIG egg.
A US cup that you use seems to be 127 grams (I had to look that up). That means when you use smaller eggs, the ratio isn't right. You may want to weigh the flour and buy bigger eggs?
You can tweak the shortage of egg by adding a teaspoon of olive oil before mixing and/or adding a teaspoon of water as soon as you feel the mixture is too dry. You can also play with it. This week I watched Giorgio Locatelli make a more elastic dough from 250 grams of flour 1 whole egg and 3 eggyolks. This is no more than changing the 1 egg ratio into 2 eggyolks. Perfect for ravioli and so.
In fact, the dough has to feel "workable" in the beginning, something you immediately feel when you start working the dough. Then you need to knead the dough with the lower parts of your hands for around 10 minutes. If the dough stays sticky, you need to add some extra flour.
After the kneading, the dough will have turned much tougher, but no worry. Pack it in foil and let it rest in the fridge.
When using a cup, I would use 1 cup + 1 big egg + 1 teaspoon of oil + 1 teaspoon of water to make it workable, if needed
Third, the rolling. I start by rolling the dough with a rolling pin! Trying to feed a slightly flattened cold ball of dough through a pasta machine will kill it soon. I know from experience.
Dust your working space with some flour and keep on dusting the dough while feeding it through the pasta machine, from time to time. You may want to cut the dough first in 2 pieces (or more) to be able to work easier.
In the very first stage, after using the rolling pin, roll the dough through the widest position of the rolls and when it comes out, fold it double, push a little with your hands to flatten and roll again. Repeat this a few times, but try to turn the dough in another direction also; you may need to fold in 3 superposed parts to do this. All this works the gluten.
Then roll the dough through all the steps on your pasta machine.
BTW, you don't need to change the ratio 100 gram flour/1egg when making pasta verde. You need around 20% spinach compared to the weight of the flour you used. The spinach has to be blanched for a minute in boiling water, drained and cooled under cold tapwater, squeeze all the liquid out with both hands until you have some sort of a spinach ball, cut and blend.