Honestly, the best thing you can do is grw a bit of a thick skin about appearing rude (sorry!). The very best way to tackle this is to ensure your dog is always kept as a distance she feels comfortbale about - which means crossing roads, ducking behind cars, doing a u-turn when you see someone, heading into bushes etc.
Once you accept that your role is to keep you dog at a distance she feels comfortable, regardless of how silly or rude you might seem, it gets better 
This is the best way for her to see that other dogs are not threatening.
If you take her too close then she will feel fear which is unpleasant for her. This will reinforce the idea that other dogs make her world worse.
If you keep her at a distance then she doesn't feel afraid and has a chance to learn that other dogs don't really change her world at all. This is the first step.
You can also pair the appearance of other dogs with treats. This will encourage her to see that other dogs make her world better. Be sure to offer/show/give the treat AFTER she has seen the dog, not before. If you do it before you teach her to mistrust food bcause food leads to dogs, not dogs leads to food.
Be very careful about tightening your lead. I know you say you don't but it is almost impossible not to. The other advantage of distance is that you will also feel relaxed which will help her. If you are too close and you start to feel nervous, your behaviour, tone of voice, even smell may change - it is too hard to fool a dog so better not to try.
As pp have said, over time and MANY safe encounters like this you are probably going to find you can get closer and closer to the other dog without a reaction. It takes time though so don't rush her. Everytime she reacts she is taking a step back so try not to put her in a siutation where she feels she has to.
Be careful about asking for a sit near other dogs. She is reactive only when 'trapped' on a lead so forcing her to sit still may well also feel like further restriction on her freedom to get away and make her worse, not better.
If you absolutely have to pass close to another dog, do it quickly and calmly. Don't stop for hellos, just move passed as quickly as possible.