Welcome to Pet Talk - Madison's mom!
Madison sounds like a little ball of fur that thinks she is a Big German Shepherd! You are doing the right thing by recognizing behaviors that need attention and trying to solve them.
I would imagine that your little one is having difficulty because of inadequate socialization when she was smaller. People tend to think that puppy kindergarten is important because you get a head start on obedience training your pup. That is not the benefit of puppy K. It is SOCIALIZATION or in another term, teaching your dog to "speak dog."
Your pup was probably taken out from it's litter before 8 weeks of age too! This is another place where learning to "speak dog" is critical. Puppies don't need to stay with the litter until 8 weeks because of MOM, but because of the littermates. In the whelping box, littermates are learning critical social skills from each other. They are learning to "speak dog" to one another.
In short, your pup is showing an inability to "speak dog" with other dogs. Don't despair though because it is not to late to work on this issue.
You need to learn a series of desensitizing steps to help your pup when she is around other dogs. She needs to get her stress level down when around other dogs and focus more on you. It will take work on your part but can be done.
Here are some steps to work on:
1. Next time you are at the dog park, do NOT allow her to interact with any other dog. Do NOT even think of taking her off the lead. Try to maintain her focus on you. Give her food rewards and praise for maintaining focus on you. Obedience work is good here since in obedience the dog learns to focus on you for commands. If you don't do obedience, I recommend you take a class. At this age though you want to go straight into a regular class and not a puppy K. class. You have lost the time period of a puppy K class being able to help socialize your pup.
2. While you are at the dog park, notice when your pup seems to lose interest in you and what body language she displays. If you can learn to "READ" your dogs body language so you can intervene appropriately before aggressive behavior is displayed, you are ahead of the game.
3. Continue to lower your dogs stress level around other dogs. Give her a massage or rub her neck. Talk soothing to her and let her know she is okay. This is very important for both owner and pup. Sometimes when you are "worried" that your pup will bark and attack, you send signals of tension and anxiety. If at anytime during your visit to the park and her levels of arousal become to intense, come back to the massage and soothing talk.
4. Next comes the harder part. Teach her a command such as "OFF" or "Leave it" or whatever you feel comfortable saying. This command is to be used ONLY when your dog gives a aggressive inclination to another dog. When focus is returned to you then you reinforce vigourosly with a food treat and verbal praise. Give LOTS of treats if she glances at another dog but does not show signs of "ruffled feathers". The purpose of this is to teach them that the presence of other dogs is a goog thing. They get lots of treats when other dogs are around!
END OF DOG PARK VISIT NUMBER 1. Make sure you leave on a successful note.
NEXT VISIT(s)!
1. Start off with the relaxing massage and soothing verbal tones.
2. Practice the controlled looking with food treats and verbal praise for wanted behavior. If you have problems in this area it may be because dogs look more appealing than your food treat. Make sure you have something that she just "CAN't RESIST!"
3. When this is going successfully, ON LEAD take her for a small walk around the park. When you pass by another dog talk soothing. Keep trying to get her attention onto you. If she does then lots of praise and lots of treats! Reward her vigorously for wanted behavior. Remember the OFF command if she doesn't want to pay attention to you. Each time she hears this command and listens to it, you need to reinforce it immediately with your food and verbal rewards!
4. Always leave the park on a positive note.
FINAL TRAINING STAGE (Only you can determine when this happens, but DO NOT rush it. Make sure the wanted behaviors are consistent first).
1. Start the visit to the park with the relaxation exercise of massage and soothing verbal tones.
2. Take a stroll and when her body language does not indicate any anxiety and she is comfortable with you and doesn't concentrate on the other dogs, try to let her off lead. Again, don't be in a hurry to get to this stage.
3. If she has problems off lead, quickly give her your command of "OFF" and put her back on lead but do not give and further verbal repremand. Just start the exercises in order again. Eventually she will respond.
Final result should be a pup that is bonded to her owner, food motivated, responsive to commands, and relaxed around other dogs! This may take weeks or months to achieve but with consistent work on your part, is achievable!
Good Luck and let us know how it goes!