I recognise exactly the sensation you're describing, and as you've said, it sounds like the start of a panic attack; before it gets to the 'full blown' stage.
I get panicky in large crowds, sometimes in big places (like supermarkets) with lots of people around but my worst ones are when I'm driving on the motorway - which is terrifying, as my 'inner monologue' starts trying to convince me that I'm going to crash the car and kill everyone on board! I also know what you mean by 'feeling detached'; I experience 'depersonalisation', which for me manifests as seeing my hands on the steering wheel and not being able to associate them as belonging to me (therefore, as I am told by the voice in my head: 'The car is even more likely to crash since there are a random pair of hands that don't belong to you steering the bloody thing!') - it feels really strange.
I know that the sensation can be really physical - the first time I had a panic attack, I was utterly convinced I was having a heart attack (despite only being 19 at the time), with a racing heart, chest pain, dizziness and feeling like I was going to faint, so I totally understand why you want to put your mind at rest.
I'd just go and speak to your GP about it - they may run a couple of tests to confirm that it's not physical, just in case (e.g. perhaps a quick blood test to check your dizziness isn't caused by anaemia) but they'll also be able to help you decide how to treat your anxiety. CBT seems like a pretty widely recommended treatment these days but it wasn't around when I was first diagnosed, so I don't know an awful lot about it, but your GP may suggest giving it a go and mindfulness is also popular. I was prescribed Propranolol, which is a beta-blocker, so it works on slowing your heart rate down and reducing your blood pressure so that the physical effects of panic aren't so pronounced. For me, it means that although my head might start telling me 'You're about to faint at the wheel, steer into the central reservation on the M5, causing a 15 car pile up and killing 22' (it's oddly specific), my body can't react by making my heart race, shooting my blood pressure up and making me see stars. I take one before I embark on any car journey longer than an hour, or where I will be driving on the motorway. I still get 'the voice', but I can shout it down because I don't get to the stage where my body is picking up on my brain's 'panic cues', if you see what I mean? Meds may not be the answer for you, but it may be worth asking about them to have in reserve whilst you wait for any 'talking therapy' to kick in. Sometimes it just helps me to know that I've got some in my handbag, just in case.
Panic attacks are horrible, I hope you get it sorted quickly - good luck!