Chase might approve you, but with your recent additions, you might end up with a limit that's lower than what you'd like. I'd let your new cards reach at least six months of age, preferably nine or twelve. Because you'll want to app for Chase cards first (5/24), that'll delay applying for other cards. In the time that elapses, you might find that the USAA card you really want is within reach.
In the meantime, concentrate on your current cards and any spending bonuses you might need to reach. And definitely plan on gearing up for another AMEX CLI in six months.
I think if you have a financial plan, then you should follow it. You said you could use the CCs you're thinking of apping for, or at least maximize your INQs, so I don't see why you shouldn't go for it. Especially with the holidays coming up, I suspect you'd earn quite a few rewards with the Amazon Prime CC. When people are apping just for the sake of apping, then yes, that's probably when they should go garden. But I wouldn't go garden if you have a plan in place and can benefit financially from your apping. I can't speak on the Limitless, but maybe do some research and see if your credit profile is good enough to be approved, but I'd definitely app for the AP and F from Chase. You're AAoA will take another hit, but since you already have your auto loan, and I don't suspect you'll be buying a house in the near future, I think by the time you need to app for anything it'll have rebounded nicely.
$500 $1,000 $1,300 $1,500 $1,850 $3,400 $3,500 $9,500 $1,000