Congratulations 
My second pregnancy, when I was 28, was exactly as you described - absolute exhaustion, ridiculous morning sickness, the heaviness of the stomach, and a niggling little feeling in the back of my brain. I tested very early on - I think it was round about the two week mark - via the GP as a routine "do you think you might be pregnant?" gateway (I needed a repeat prescription and she took one look at me, listened to the knackered whinge, and insisted on testing me), and the nurse was astonished when the line on their test strip came up immediately. I knew when conception had happened, because it was the only time that year that my ex had managed to convince me to DTD (February 14th...), and was asked, almost immediately "do twins run in your family?". Well, yes; they do. My grandmother was a twin (her twin sister was stillborn), and had twins (although my uncle's sister died shortly after birth). My mother also miscarried twins between myself and DB2. It wasn't out of the realm of possibility.
At 8 weeks, I started to bleed in the middle of the night. Had to wait five days for a scan, due to the Easter weekend that year. Saw the tiny little bean that became my DS (12) and what was left of his twin. The radiographer was brilliant. The doctor she called to talk to me? Not so much. But I have a wonderful, funny, kind, loving 12 year old son and, frankly, although I'll always wonder what his twin would have been like (I like to think it was a girl), I am so grateful that he hung on in there. Even if I did spend the entire rest of my pregnancy absolutely terrified and swore I would never go through another one because of the whole experience! (I also wonder if, someone had seen me that night, rather than five days later, if his twin might have been here, too).
Twins aren't just a "thing" of pregnancy in older women/intervention. They also run in families. Trust your instincts, OP, and again - congratulations, whether it's a single or a multiple!