You are right that your overall fitness may suffer in OSUT if you don't put in extra work. Your PT score will be fine(because that's what you're trained on) but other areas, most notably overall strength may lag.
YMMV a lot on what kind of equipment you'll have available to you and what you'll be allowed to do on your off time. When I went through one platoon had weights, a barbell, and a little bench set-up in their barracks room because their DS were into that. Their head DS was from Ranger Bat so he knew a lot more about proper fitness then the other DS. They would routinely do extra hill sprints, calisthenics, and work on the pull-up and dip bars in the afternoons on the PT field(supervised by their DS). We didn't do much of that. We just got smoked to hell and back. A rather less effective training system.
A big part of keeping up your fitness will be up to you to put out during PT. I went through in 2009 but I imagine it would be pretty similar - in that you'll have runs based on ability level, so those that are fastest will run with the fast group, etc. Then you'll do a lot of timed sprints so again you'll be able to put out 100% and get good results. YMMV on how many pull-ups you get to do. I remember we didn't do much in the first 4 weeks of OSUT but after that our DS made us do 10 pull-ups everytime we went to chow, so I'd imagine you'd get some sort of opportunity to do more pull-ups as you get into your later phases. Push-ups and sit-ups you can knock out more of in your own spare time. Plus you'll get to do plenty when you're getting smoked, so just make it a point to do really good proper form push-ups whenever you get smoked rather than sandbagging them like most people will.
In the end you're going to lose some strength if you're used to weight-lifting. There's nothing you can do about that so just don't worry about it and focus on doing what you can do - loads of push-ups, improving your run even more, and just generally conditioning your body to a higher overall volume. I'm reading Sua Sponte by Dick Couch talking about 75th Ranger training. He said something really interesting, that they're trying to train for durability. Not specifically strength, or endurance, or speed, but durability. I believe the quote in the book was "can you walk all night and then still have the mental and physical stamina to raid a house in the morning." Much of athletic training is structured so that you can have peak performance for an hour or two. Military training will be much more focused on getting you to be able to perform hour after hour, day after day.
Oh I just thought of one area I would recommend - loading up your ruck and walking the stairs in your barracks. I did this toward the end of my OSUT. When I went through we really didn't ruck that much. Maybe we'd go once a week, and it was always pretty light and pretty short. I don't remember ever doing more then like a 6 or 8 mile ruck until the final FTX with the 12 mile. Since rucking is going to be a huge part of selection I'd throw on a ruck and walk those stairs in your barracks. Get your body more conditioned to having a heavy ruck on your back.
Good luck! Don't worry too much. Just work hard, do what extra you can(without injuring yourself or overtraining), eat like a monster, and all you gotta do is make it to the next chow time.
