Family history - stomach cancer

Hi,  I hope it's ok to post here, I did posti n signs and symptoms but not sure if this is more the place.  I'm 37 and for about 2 years I've been having issues with heartburn, bloating and feeling full very quickly.  

Recently I feel like things have been getting worse, I'm getting these symptoms:

- heartburn

- pain when swallowing- particularly things like chips and cold fizzy drinks are very painful

- bloating

- a pain around my sternum and centre of my back

- mild pain across the top of my stomach

- pain if I drink alcohol (rarely) 

- if I take rennies or something like that the heartburn eases but I get pain across my stomach instead

- niggling pain under my right rib just below my breast area. 

- very occasionally I get a spasm type pain in my chest which feels like a heart attack but goes away if I drink cold water- I assume it's indigestion.

- Palptiations- probably not related but just including as it's something new I've experienced in the last 2 weeks. 

I'm only 37 so I know it's unlikely but my mum passed away with stomach cancer in her early 60's and her gran also died of stomach cancer so I'm concerned there is a family history.  

I've spoken to my GP and he's recommended trying omeprazole for 2 weeks and if this works then he wouldn't have any concerns, however I've read a lot about symptoms being masked and I'm not sure if i should push for a referral anyway due to the history and symptoms?