I, too, adopted a fearful, cautious, shy dog named Gracie who I write about a lot here on this forum.
I have worked extensively with her to help her master virtually all of her fears-- and be way more confident in life. It has taken a lot of time and patience and genuine understanding of her and her mindset. And lots of gentle, humane training and tons of counter conditioning! To teach your new dog Tulip to walk on a leash, I would suggest getting the highest value food that your dog will work for!!
I know you said you have tried high value treats, but sometimes it is just a matter of finding the highest value that your dog absolutely loves. The reward has to be greater than the risk!!
I have seen this with tons of dogs of every personality/type and it is all the same. The food has to be enticing enough (amazing tasting and smelling!!) for the dog to be brave enough to try something new or scary.
I can't tell you how many times people tell me their dog won't take treats at all, and lo and behold I have their dog not only taking treats from me, but doing fun tricks and climbing on my lap to get them. Yes, even the super uber shy doggies. But I make it very fun and ultra rewarding all the time when working with these dogs. Sometimes I stay very quiet or whisper and that really seems to help with shyer/timid dogs.
Anyway examples of high value food I use with Gracie or other shy dogs:
1)Home cooked chicken
2)Rotisserie chicken (just used this today for thunder training with Puma pup!)
3)Cooked chicken gizzards/livers
5)Warm Meat bites (just used this all week with Puma pup to teach her that having her ears cleaned gets her yummy cooked meat bites!...Also I put the meat juice over some kibble for this between steak bites)
8)Melted cheese on tortilla
9)Hot dogs--crappy, but they do smell great to doggies! Great for reactivity training as you can toss them and the dog can find them easily!!
10)Fajita meat/chicken (Gracie loooved this at the vet when I gave her this for getting her scary shot!)
12)Peanut butter on a cracker, or let dog lick it from your finger, or a tupperware
13)Very, very small amounts of ice cream (Puma gets this for letting me put liquid bandage on her head scrape)
14)Macaroni and cheese bits
17) And more! My doggie treat pouch is always with me when I am working with my dogs! I love to reward all good behavior with genuine praise and yummy treats as needed.
So some of these things I listed are not so healthy, right? My motto: It is far better to give my dogs small amounts of "unhealthy" or imperfect food at times as I teach them to conquer their fears and gain healthy confidence, especially in the beginning!
So, for Tulip, I would load a treat pouch (or fanny pack) with tons of very small amazing food bites in some ziplocs in your pouch. Experiment before hand and see what she really, really loves to eat. Warm it up first, great smell has everything to do with risk taking for dogs, right? Tip
: I teach "Gets it!" first so my dogs all know to look down on the ground for the small food bites when I say "gets it!"--this really, really helps! Plus is great distractor of fear/uncertainty.
So... take Tulip outside on a leash, toss the amazing food bites on the ground in front of you. Say in a gentle, happy voice, "Tulip, gets it!" and then point to the food bites on the ground. Keep gently tossing the food bites in front of you so she will walk forward. Don't be shy to use tons of food for this as you really want her to be able to walk confidently on a leash, which is needed for her safety, esp if you will be traveling soon.
If she is hesitant to walk forward in the beginning, then just have her on the leash and toss the food on the ground and have her walk to it at any angle or direction. The point is to teach her brain that walking on a leash really, really pays or is ultra rewarding!
Also, I personally find it best to do this type of training/brain conditioning when my dogs are a bit hungry or before a meal.
The food will be more enticing on an empty tummy and they will usually be more willing to try harder or be braver to get the yummy food.
Like this morning, I had to put liquid bandage on Puma's head for her head scratch. And medicate her ears with some goey mineral oil. She really, really wanted to skip the whole process and go hang out with her squirrel buddies in our backyard.
So, before she ate her breakfast, (and before beloved squirrel time) I did the whole medication thing. Not her favorite thing. But we are getting so much better at it together. She was originally snapping at me (out of fear) when I would touch her ears with the goo. I understood completely as this was all new to her and pretty creepy! I would hate someone pouring goo into my ears!
So...I warmed up a bunch of steak bites, cut into pea size bits, poured the warm meat juice over kibble and hand fed her all these pieces as I gently and slowly medicated her ears. This time she even allowed me to wipe her ears out with a tissue without snapping. Next I am working up to using a q-tip on her ears, so today I traded her steak bites for letting me gently touch her face with the q-tip to show her that the "odd item" isn't hurtful. Very, very good session today for both of us. She loooved the yummy steak bites--me, too! Sorry so long---I just know from experience that sometimes finding just the right food for the individual dog can really make a huge difference in training, especially with timid or cautious/fearful/shy dogs!