I won’t delve into the reasons you shouldn’t based on ‘coolness’, but…

There are half-ways between the pronunciation you’re mentioning: the short ‘a’ sound can be used (common in British English), and wouldn’t seem so jarring to me.

You can drop your Ts for weak Ds in things like ‘water’, so the tip of your tongue gently taps the roof of your mouth instead of pronouncing the typical British hard T (a lot of us don’t pronounce it like that, mind you).

However, “speak like an American” doesn’t necessarily mean one accent, as there are many in the US. Choose one, I suppose.