Yep, I still feel awkward at the gym sometimes. It helps that a lot of older people and bigger people go to my gym, rather than trendy slim "gym bunnies"...
In classes, most instructors are great at giving high impact and low impact options.. the key is to keep moving! So if you need to stop and grab a drink, just walk on the spot as you go..
Also, maybe talk to the instructor before the class, say you haven't really done much exercise in the past and don't have a lot of cardiovascular fitness, they can usually reassure you and tell you its okay to take a break if you need it, and maybe show you some lower intensity options..
In the end, remember WHY you are doing this - you are becoming a fitter, healthier you! And everytime you leave the gym after a workout, you are one step closer to that goal.. And with each class, your fitness will increase, and you'll be closer to the day that you can do it without having extra breaks!
Stay motivated, keep it up, I know you can do it..