the A1c is not an accurate number for BG control. it is based on the the amount of sugar stuck to red blood cells and the assumption that red blood cells live 90 days. if YOUR red blood cells live longer then your A1c will be higher.
BUT if your A1c is accurate and your fasting numbers are good then your post meal numbers need to be high to bring up the BG average for a higher A1c
Hba1c of 6.7 is an average blood sugar of about 160 (some different calculation method may say 150). This is already diabetic. From your history, this has clearly been developing over the past 4 years. It is a shame your doctor hasn't discussed the implications of this obvious trend over the past few years.
Fasting blood sugar should ideally be 70 - 80 range.
Non-diabetic HBa1c is < 5.0 (and research has shown that once hba1c exceeds 5 there is a good chance of it getting higher over time if action is not taken).
You need to discuss with your dr regarding medications (maybe metformin).
If you are overweight, try to normalize weight. Daily exercise please. Look into low carb, moderate protein, high (healthy fats) diet.