These days, especially if you choose to run alone, you should carry your phone or some communication device with you at all times - regardless of if you plan to talk much on it or expect any calls. It is simply a safety precaution that may be the difference in you being able to dial for help in the time of an urgent need and being left stranded, alone without a way to contact or communicate with anyone. And fortunately, there are a lot of options these days to carry your phone while running!
You can always carry your phone with you in your hand. But most runners don't prefer this because it's annoying, inconvenient, and difficult - especially if you're trying to keep your hands relaxed. The next step up would be to secure your phone in your sock, shoe, or under your shoelaces so that it's strapped in. Still, that's not always the safest option as it will likely slip out.
The first option that is safe is to buy a carrying device specifically for your phone. A lot of companies produce armbands, belts, or strap ons that easily fit today's sized smartphones (which will also fit anything smaller). A lot of folks use their cell phones as music playing devices as well while they run, so you can also look for bands and belts that are meant for cell phones or mp3 players. Usually, they are made from durable material that can stand up to sweat and moisture while still protecting the phone, and they often stretch to fit around multiple sized phones.
The next option is to purchase running attire that has special pockets to in which to store phones. You said that you hate the bounciness from a phone flopping around in your pocket while you run, and that makes sense. A lot of running clothes companies make shorts and tights, as well as long-sleeved shirts, that have hidden zip pockets that are fitted tightly against your body to reduce bounce. Just zip them in, and they will stay put!
Тhese are some of the ways that explain how to carry phone while running, but still you should be very careful.