Try picking them at different stages and you'll find the perfect moment when the peas are filled out enough to be worth picking, sweet, and not yet gone starchy. Some like them a little younger, some like them a little more substantial.

They will, however, taste like peas rather than like candy. :-)

I grew Sugar Snap vines for the first time this year. I found the pods were the sweetest when allowed to plump up quite a bit. Just as long as the peas inside weren't actually firming up and getting starchy. I found them much better eating like this.

Mine didn't seem as sweet this year as I remember from last, but I also found that if you let them plump up a bit more, they do sweeten. Downside was (on mine at least) they formed more of a string while they plumped.

Here is a link that might be useful: Tales of a Transplanted Gardener

Are you sure you did not grow snow peas? your description matches that of snow peas, which remain flat. If they are, delicious stir fried and I also toss them raw slivered in salads. They are not sweet, but still good.