Chest Pain / Handlebar Width?

I have chest-shoulder and trap pain, if I punish my pectorals-shoulders the day before riding.

I have some triceps tenderness, whenever I do lot of dips before riding.

I often have some upper back tenderness, when I do more than 100 pullups and suspension trainer rows the day before.

Oh, and I can get nasty hamstring-glute tenderness, when I do too much lunges.

Chest pain from angina is different, like crushing pain, often radiating to the left shoulder-upper arm. 

On the other hand, if I'm not sore, I can spend whole day in the saddle without numbness-tingling-funny feelings. I'd do some general strength training with suspension trainer-core excersizes-pushups-pullups-rows-dips, stretch, and even take a look at bike fitting.

Ur hands are not for support. they are for handling-cornering, but not for weight support  

GO AND SEE A DOCTOR. Please.

It might be postural, or it might be a sign of incipient heart attack. Seriously, do this.

I'd also check bike position, having too much weight on your hands (often due to saddle too far forwards or tilted too far downwards) creates extra work for your upper body. Doing some light weight training (deadlifts and squats) with good technique can also help develop back strength which also will help take weight off your hands and keep you stable.

as others have said, developing all around strength is never going to be a bad thing.

Cycling is a wonderful form of excercise but long rides in the same position can and do expose other physical weakness's.  Cross training, like moderate weight training, stretching, core strength work wonders for better bike posture.  I do  basic compound exercises, like chin ups, dips and squats to keep overall fitness.

 Move around, change hand position and get out of the saddle when riding.  Dont keep the same position for too long or stress injuries can occur.  

It's amazing what you can wrench sometimes. Managed to strain my triceps from having a death grip and pulling myself about, instead of relaxing on the bars and using my core to get in and out of the saddle.  

Have a shot at building a little strength in the chest before splurging out on new gear I reckon. Just do some half pushups for a bit, girly ones, then work your way towards full ones. Two or three times a week. 

PITA to do stuff like that, but sometimes you do need to do a little strength work if an area starts playing up and chest is pretty easy to target..

Won't always work, but it's free so first port of call. 

(FWIW, I have to do crappy indian knot and cat stretches few times a week. Hate it but if I slack off even for a week my ITB and back pain starts to creep in again).

How many proper pushups can you do in one go? Truthfully.

What sort of terrain have you been riding? Have you been tackling stuff where you've been getting out of the saddle and maybe keeping too much tension in your upper body?

Are you stretching before riding?

I'm having to do 15/20 minutes with my advancing years, insead of just jumping on a bike as in my youth.

Here's a quick Art's Cyclery video link on determining handlebar sizing

https://youtu.be/clskb8I9-j0

Do you have a  sports medicine doctor in your area who's an avid cyclist? He/she may understand the cause of your chest pain from a medical point of view. I'm no doctor by any means but one particular would be Pericarditis, inflammation of the sac surrounding the heart.