To actually test for whether they are pregnant or not, you can send off a blood sample to Biotracking. They can also test for CAE, btw, so it is a good way to tell if your herd is clear or not. I draw my own blood, personally, so I don't need a vet for that.
To tell how far along they are, well, that is a lot trickier. The only way to know for sure is a vet doing an ultrasound.
The larger tummies might not be a sign of them being pregnant at all. You own cows, so you know how rumens work. Sometimes when a goat fills up on a lot of browse and hay, their rumens are nice and round. Not bloated, since everything is normal.
If it were me, I'd actually be drying up the Saanan now. Her body condition is rather poor, she's on the thinner side. Heavy milkers like her put all they eat into the milk bucket. Giving her time to be dry and put some weight back on for a healthy pregnancy and new lactaction cycle. Drying up 2-3 months into their pregnancy isn't enough to really help them put that weight back on, when they start out thin.