Hi there I would go to gp blood in saliva could be due to chest infection,coughing too much thus bursting tiny vessels in throat(not serious) if you have severe gastric disorder it could be coming from gastric is the blood bright red? If so this is prob due to strain from coughing,is the blood brown in colour if so this is old blood and should be checked out ASAP are you on medications that could be making gums bleed thus leaving slight blood in your salvia.
it could be an ulcer see gp
Is the blood seen when coughing up phlegm and mixed with saliva? Do you have any bad teeth or gums, that could be the cause of it? If it's mixed with phlegm and you see blood, you should definitely go to your doctor ASAP (which you should do anyways with the gastric issues going on). If it's not associated with coughing, it is most likely related to teeth or gum issues and you should get seen by your dentist. Bleeding gums is a sign of poor dental hygiene/gingivitis and it's easily treatable before it gets too bad. Good luck!