Yes, it is often possible to correct posture with proper posture exercises at home. Whether you can do it with exercises alone depends on specific postural issues and their stage of development.
When trying to correct your posture this way, you have to make sure your posture is improving. you need to regularly evaluate your progress. You have to know if exercises are working or if they actually hurt you in some areas.
Thus, you should regularly check your posture with a doctor. In addition to this, you can use specialized apps, like our app APECS. APECS analyzes several photos of you from different angles, and gives you a precise table of your postural angles. So when exercising at home, you can regularly verify your angles and track the improvement.
Overall, posture correction should use a global approach. It should include sessions with a doctor, posture exercises at home, proper diet, sleep rhythm etc. When doing posture exercises at home, you have to make sure you are making a positive impact on your posture. Such posture tracking methods like APECS could help you with your goal.