You have to keep in mind that the rating of the trailer is based on the rating of the axle. 1-3500lb axle gives you a rating of 3500lbs, 2-2500lb axles gives you a rating of 5000lbs. Most dual axle rigs will put you over 4500lbs, you'll be overweight. You are almost entirely limited to single axle. There are a few exceptions like livin lite, you could get the Camplite 21bhs or 21rbs which will give you exactly everything you want, but you'll have to be careful how you load it.
You could get a trailer manor, with it's low profile, i wouldn't worry too much about sway. No walk around bed.
You could get an escape trailer as mentioned before.
You'll also need a weight distribution hitch. For any box trailer you'll be towing with a pilot. I can't believe YKCAMOM drove without one. Could have been seriously hurt if a large wind gust came around. If you are using a marginally adequate tow vehicle, i'd get a hensley arrow or propride hitch for sure. These hitches create a virtual pivot point closer to the rear axle, you'll be far more stable like a fifth wheel.
However, none of these options really keep you under $10k and give you a walk around bed. If you forgo the walkaround bed, you can probably get a small 16 foot in your weight range that has double twin beds so you can walk in the middle. If you are a handy kind of person, you can find some late 90's airstreams with double axles with twin beds for your price range. You would probably be limited to the 19 foot. Even then, you'll be pushing your weight limit hard. Airstreams are more aerodynamic, especially from side winds which helps immensely with sway.
I can't find a twin bed model, airstream calls those the A models. Here's a late 90's bambi who's GVWR is right at 4500lbs. Dry weight at 3500lbs. But it's a double bed.