Hi Hannah, and welcome to Talk Budgies! You will find lots of great information here about caring for your budgies.
Congratulations on bringing home your two new friends! Your budgies are going through a huge adjustment period, and it's normal that they are stressed right now.
What kind of cage have they got and where is it located in your house? What else goes on around their cage? You can put a blanket or sheet over 3 sides of their cage to help them feel more secure right now while they settle in. Also make sure that their cage doesn't sit too close to the kitchen or bathroom and that it isn't near an air current (like a vent).
Some budgies get diarrhea from coming home due to the stress. The green ring that you're describing can mean several things, one of which is that they aren't getting enough nutrition right now. This could be because they are stressed, or it could be because of illness.
If I were you, I would watch them closely for the next day (but try to leave them to settle in; you can play relaxing music or budgie noises, but I wouldn't try to "bond" with them right now). If you see that they are puffed up, vomiting, sleeping a lot or hanging out only on the bottom of the cage, you will need to take them to the vet. However, they might get over their initial stress and feel much better in a few days.
You can also read about your budgie's diet on this forum. It's great to offer them fruit and veggies every day so that they get their nutrition. Most budgie owners also recommend using probiotics. I use Harrison's Booster, but other owners have great success with other brands.
Best of luck, and please let us know how you're getting on. We'd love to see pictures once your budgies are settled in!