FOOD COOKED WITH MEAT

Question

I WAS WONDERING WHAT THE HALACHA IS REGAURDING EATING FOOD COOKED WITH MEAT AND AFTERWORD HAVING DAIRY. I LEARNED THAT IF I ONLY EAT A POTATOE FROM A CHOLENT THAT HAS MEAT IN IT, ALL I HAVE TO DO IS WASH MY HANDS AND RINSE MY MOUTH AND THEN I COULD HAVE DAIRY. DOES THIS ALSO APPLY TO FALAFEL BALLS OR FRIES THAT WERE FRIED IN THE SAME OIL THAT MEAT WAS FRIED IN? PLEASE LET ME KNOW. THANKS FOR EVERYTHING