You've been around long enough here to know how some rescues generally feel about taking on aggressive dogs, so I think you need to be very realistic about what you're asking. Dog-aggression prevents many dogs from entering foster-based rescues, as foster families nearly all have personal dogs that they want to keep safe. Many (but not all) breed rescues also try to avoid mixes, as they're harder to adopt out if the people who are coming to breed rescue to adopt are looking for purebred-looking dogs.
I'm sure that you're also aware that good rescues in your area are inundated with dogs -- Houston is a high-kill city, and last I heard, the GSRs there had more dogs needing help in shelters than available foster space. Many of those shelter dogs have no issues -- dog friendly, people friendly, no problems at all, and will be put down for lack of space. Dog-aggressive owner-surrendered dogs in my city get walked straight from the drop-off room to the euthanasia room -- they're not even put up for adoption.
If you contact rescues, your best chance is probably to (a) ensure your dog is fully up to date on vetting (and neutered), so you're not asking them to pay for neutering, and (b) offer to foster your own dog
until a home is found by the rescue. Many of us don't like doing that though because at the last minute, people tend to get cold feet and decide to keep their dog, after we invest months finding a great home--this happens literally on adoption day when there's a lovely new family waiting to welcome the dog, and the owner suddenly realizes they can't bear to part with the dog after all. So don't be surprised if they decline your offer, and don't take it personally -- it's because some rescues have been burned by investing a lot of time into helping rehome dogs like this in the past.
Instead of burdening overtaxed rescues in your area, I recommend investing your own time to try to find a home through Rehome by Adoptapet:
You can list the dog on the website free of charge, right next to the shelter and rescue pets. It generates an adoption app and adoption contract. (The rehoming fee gets donated, so that nobody can sell dogs on the site.)
Dogs that must live as only-dogs take longer to find homes for. Most people don't want to manage that. Try to plan to start looking for a new home a few months before your move, as this kind of dog (a mix with issues) is likely to take longer to place in a good home. It can be done -- emphasize his good points in his bio but be honest about his issues, and what kind of home is needed for him.