Is your new window air conditioner the same BTU output rating as your old one?
An air conditioner sized too small will run constantly and an air conditioner sized too big will shut off too quick and not remove the humidity making your house feel clammy.
If your room is hot where you installed the air conditioner and it's been off for awhile. It may take your air conditioner some time to bring the temperature down.
If it still doesn't shut off then your ac might be too small. The right sized one should run for a long time and then shut off periodically.
But one that shuts down too soon will be too big.