I am so glad you are making progress with your horse.
Contact is a matter of trust to the horse, their trust that you won’t try to choke them with too hard contact, and trust that you will give with the reins a little bit when they obey the slowing down or stopping hand aids.
When I work with my lesson stable’s horses who are reluctant to accept contact I make sure to keep my fingers relaxed and supple.
Also, when the horse’s head is moving forward, I give a little bit more with my relaxed fingers when my hand is the furthest forward. I only give halting or slowing down hand aids when the horse’s head is moving BACK (really subtle at the trot), and when his head moves forward again my fingers are relaxed and giving a little bit more to encourage the horse to reach just a fraction of an inch more forward with his relaxed mouth.
At one 1/2 hour lesson a week it generally takes me 4-6 months to reliably get relaxed happy contact that the horses willingly reach for. These horses were either sucking back, head flinging, pulling my seat out of the saddle as their head went down at a high speed, trying to run away (at a walk and trot ha, ha) and showing great discomfort with the bit and contact.
I have found that a lot of horses LIKE kind sensitive hands with relaxed fingers. Of course, horses being horses, others will prefer firmer contact.
You are making progress. Just remember, even with the vet work and the good training it will take TIME to get everything good and reliable.