Have you taken it to a different store? If you haven’t try it at another place. If it’s only a problem at one store (especially if it’s just on one of their machines), it could be a problem with the merchant. It could also be a problem with the mag strip or chip. You can ask them to key it in, that would tell you whether or not it’s a physical problem with the card as opposed to something happening after it leaves the machine. Also, you said debit, often times if a debit card declines, it’ll work when you use it as credit (no PIN). Still something you’d have to figure out, but it’s always worth asking the cashier.
If it doesn’t work at multiple [different] businesses, especially if they’re different types of businesses (gas station, grocery store, Best Buy etc), then there’s something up with your account. For one reason or another it’s locked out.
Since they sent you a new card but didn’t suggest anything wrong with your account, it sounds like they think there’s an issue with the physical card. Again, bad mag stripe or chip. One thing that you can do if you have to call again, is to ask them if they can see it hitting/declining on their end. If the request never even made it to them, it’s likely not them. If it did, they (most likely) declined it for one reason or another.
Do you know what the machine said when it wouldn’t work? Was it simply ‘declined’ or was there any other kind of message to go with it (card type not supported, cannot read, incorrect PIN, network error etc).
One other thing, if you do ask the cashier to try something else (key it in, plastic bag trick, credit instead of debit…), try to limit it to just one other method, at least if they have a line. It’s a PITA, when a customer ties up a line for 5 minutes while they do all kinds of voodoo with their card.
ETA, I see you did use it at two stores, but they’re probably in the same category. Issuers do have the ability to block categories of merchants or even set daily/weekly/monthly limits on categories, but you usually only see that with business cards. So, you may want to try it at another place ie gas station (to keep all things equal, try buying something inside so it gets used in a terminal, not at the pump).