Can I straighten my teeth with braces at the age of 27? (Photos)

I am going to be blunt here:
Get a new dentist.  The information you received shows a real lack of understanding of the current standard of care.
Your kind of bite, along with the excessive tooth wear it can cause is a common reason for adults to seek treatment.  I have treated patients well into their 60's for similar bites.  The sooner you fix your bite, the less restorative work you will need to rebuild your teeth.  Your smile will look younger, better and healthier when you are done.

You are certainly not too old for braces. I suggest getting a second opinion from an orthodontist. You'll be so happy when they are finsihed. Best of luck.