Some cis guys don't dress to the left or right either and keep their junk in the middle, so they will probably have heard of it before. You could always make a bit of a joke out of it and say something like 'Front & centre!' when they ask how you dress. You can wear your packer loose in your underwear if you have briefs: you'll keep your underwear on throughout the fitting. Even better, wear your packer in some briefs & put a loose pair of boxers on top of those. If you don't feel like wearing your packer, just wear some loose boxers instead. They do try to avoid touching your junk when they measure you, so don't worry about them getting too much information.
As for locker rooms, I found it handy to ask the manager for a quick guided tour to familiarise myself with everything there; that helped with the anxiety & it allowed me to check that they have private shower facilities & cubicles. If they don't have private showers, you can wear your towel around your waist as you approach the shower, face the wall (away from everyone else) & take your towel off to shower - and don't turn around to face the room whilst showering. If possible, pick a shower in the corner so you have extra privacy by facing the corner. Then wrap the towel around your waist as soon as you're done.
If they don't have private changing cubicles, you'd be surprised how much privacy you actually get in open changing rooms. Most cis guys have been raised in an environment where they've internalised a whole load of homophobia; they're taught that looking at another guy's junk is 'gay' and a bad thing to do. So they'll generally not be going out of their way to look at you. Simply find as quiet a spot as possible, face away from the room, and get your undies on as quickly as you can. You can even stand in front of an open locker door to give you an extra bit of privacy: if you pick a locker at waist level in a corner of the room, you can have the corner hiding one side & the open locker door hiding the other, giving you excellent privacy. Alternatively, you can simply take your clothes to the toilet & use a cubicle there. Perhaps they even have disabled facilities that you can use, if you'd find that easier.
If you have chest scars, don't worry about it: I've displayed my (hypertrophic) chest scars openly in the changing room for about 2 years now and nobody's ever said anything. But if they did, I've prepared a bunch of cover stories, including a double lung transplant, open heart surgery, and a shark attack!
Make something up that suits you.
As a final tip, I told the gym manager that I'm trans because we have legislation that protects me from discrimination. I was worried that some other gym member would clock me & freak out... so I made sure the manager was on my side right from the start. Nobody has clocked me and nobody has freaked out.
