My symptoms seem exactly the same as yours to infinity and beyond.
Currently able to run through the discomfort but the following day my painful right hip and groin certainly restrict my movement. A further 24-48 hours later and it has cleared up until I run again.
I previously suffered with a tight IT band on my right leg, which was sorted by one physio visit and following some stretching exercises, all relatively quickly.
I think my problem is possibly associated with a back injury I suffered a few years ago (slipped disk L3/L4). When I started exercising and running last March/April I started using a personal trainer to work on strengthening my core. This did wonders for my back and I went months without any back pain (unless I did something stupid). I stopped using the personal training at the end if October and in all honesty my core exercises have ceased since. These problems started during December, previous to this my hip/groin were pain free.
I'm going to start working on my core and get back to my personal trainer in March, in the hope that my problem will clear up.
Glad you've taken the first step. Don't worry about feeling tired and out of breath. This happens to everyone when we start from scratch.
Stick to the plan, if necessary repeat certain weeks if it proves a little difficult, no shame in that.
Take your time, the important thing is that you've taken the first steps.
Couch to half marathon is possible, but you'll need to be dedicated and follow a decent plan focusing on just finishing and not a good time.
Most important build your mileage slowly, and remember rest days are just as important as running days.
As has been suggested give a local Parkrun a try, they are great fun and a great first target of achieving 5k without walking.
Keep us posted on your progress and let us help keep you motivated.