Welcome to Sclero Forums. I'm sorry that your mother and grandmother have scleroderma, and that you have new onset of finger joint inflammation and swelling. Since it is a joint issue and might represent some type of arthritis, it would be very appropriate to see a rheumatologist. It often takes months to get a first rheumatology appointment in many parts of the country, so you might as well start on it now.
Please keep in mind that your family history of scleroderma doesn't mean that you will also get it, in fact, a family history of autoimmune diseases increases the odds of acquiring any autoimmune disease and not one particular type. So, you might just as easily have gout, which is the most prevalent form of arthritis. In any case, its almost reassuring, from a scleroderma standpoint, that your joints are so swollen and the onset is so sudden, as scleroderma tends to begin slowly and insidiously, and cause the fingers to swell all over, not just in the joints, so that they look like sausages. Also, there tend to be other symptoms early on or perhaps before that, such as Raynaud's. And joint deformity is much more of a rheumatoid arthritis thing than a scleroderma thing.
Not that its better to have one particular type of arthritis rather than another! But some types are more treatable and the treatment will vary depending on the cause. If, for example, your initial tests reveal that you have gout or a clear cut case of rheumatoid arthritis, then you won't need to see a scleroderma expert. So do make a rheumatology appointment.
To re-establish your precious peace of mind for the next few weeks or months, you may want to try my favorite trick of trying to identify and then improve my self-talk. It's easy for any of us to talk our way to the top of a cliff, but we can also talk ourselves down from the cliff.
For example, we can tell ourselves that our finger "changes" are "interesting" but that "nobody ever died from swollen knuckles". Or that "millions of people around the world have arthritis, if they can live with it, so can I," or "I'm probably a lot healthier than my mother and grandmother, plus whatever is wrong, I could get appropriate medical care", or, "I'll be okay because I have good information and support right away." Even the old standby of "It is what it is" can be strangely comforting, perhaps because it doesn't give our imagination permission to go rogue into total drama mode.
Please stay in touch. We are here for you, whether you have only a temporary inflammation due to a passing thing, or whether it turns into something more than that. You'll get through this phase, and regardless of whatever develops, we'll be here for you.