There are a couple ways to do this. It's not a bad idea to wait until you are living in the house with your stuff for a while before painting. It gives you an opportunity to notice the changes in color in the room based on the sun moving across the sky.

Alternatively, you can take a couple of the major colors of your furnishings - a sofa pillow, a dining chair or buffet drawer - to the house now along with some color chips and see what you like.

I used the Ellen Kennon full spectrum paints in my house. She has a large palette of colors, but not so large that it is overwhelming.

We built a new house and I knew the builder was going to be bugging me for paint colors. While we reused a lot of furniture from the old house, most of the upholstered pieces were either neutral or were being replaced, so colors could go any way. So before the sheetrock was even up, I was picking paint colors. Fabrics speak to me, so I did it by selecting fabrics for the window treatments. The DR is open to the Library so I kept the colors in the same family though with different intensity, the Library being darker and more intense. I put the DR wall color on the ceiling in the library and the library wall color a shade darker, so they all blend. DR walls BM 2165-50 pearl harbor, library walls BM 2165-40, dark beige.

In the guest room I was inspired by many rooms in decorating magazines that used toiles. I wanted to go with that, though I didn't use an actual toile but used something similar in the yellow/blue french combination. But the fabric came first, the paint second. I used BM pale moon OC-108 on the walls, and BM cumulus cotton 2063-70 on the tray ceiling.

I had no windows in the laundry room or the powder room. When we were looking at houses to get ideas, there was one kitchen that had the off white antiqued cabinetry with heather paint on the walls and I was in love. So I put that heather color in the laundry room....BM tropical dusk 2117-40.

When I was at a museum one time, I saw William Holman Hunt's "Lady of Shalott" and was in love. I bought the poster and framed it and used the colors in there as inspiration for my crazy powder room...it's a powder room so you can go wild. People either love it or hate it. It is definitely special. Walls BM adobe orange 2171-30, ceiling soft fern 2144-40. The tile floor is a dark blue-gray like the sterling silver in the poster.

The masterbath was inspired by the tile which is marble and glass dot decorative placed in a field that mimics onyx. The glass is taupe and the marble has some deeper and paler copper shades, so the tile came first and the paint colors followed. The bath is tall 9' ceiling but narrow, so to bring the ceiling down, I painted it BM copper mountain AC-12. It blends perfectly with the Egyptian Copper corian I used on the counter and the shower surround.

Bottom line, for each room you need an inspiration piece that you love...a plate, a fabric, a pillow, a blouse, a picture, etc. and choose your colors from there. And if it is a piece you love, you will love the colors that come out of it. I say, why not let the artist who designed the piece help you with your color scheme....

It's been my experience that it's best you wait till you've lived in a house a year before you do any major decorating. You need time to see how light affects color in each room. Time to see how you really will live in the space and how you will use the space.
If it were me I'd opt for to bring in color with the accessories till I figured out what I really wanted to do. Accessories can be inexpensive and be changed out fairly quickly without killing the budget and can work well with what you already have. After such a big remodel my budget would be sagging and I wouldn't want to buy new furniture if I couldn't get what I truly loved. Using what you have with a few new things gives you time to build up your savings so you can truly make the house the home you dream of.
In the meantime I'd be going thru magazines clipping pics of things I love and putting them on a board to see where my decorating senses were sending me. You might find that you are drawn to certain colors. Then it's just a matter of figuring which room or rooms would best be served by using those colors.
Really good decorating doesn't happen in a day or even a month. It happens over time.

It's certainly possible to decorate a house before you actually move in. I would start with fabric and furniture first. Paint colors will come last. Do you have floor plans with measurements? How about inspirations pics? Houzz is perfect for that. It will depend a lot on your comfort level and thought process. Are the sort of person who debates with herself and needs to see 1000+ options and even then isn't always sure of the decision? Or do you know what you like and are able to make important decisions with confidence? Either way, you could hire a decorator or designer to help. Your architect or GC may have recommendations.
I couldn't imagine living in an incomplete project especially after living through the chaos of a remodel. I would want, at least, some of the spaces to be finished. We just finished our remodel. After many nights of sitting in what felt like a foot of drywall dust and acres of plastic sheeting, what kept me sane, was picturing the furniture, paint, rug, draperies, etc that I had already chosen in those will-one-day-be-fab rooms. They were 95% "complete" weeks before the house was ready. That worked for me, YMMV.

We too moved out of our house for a whole-house reno. For the paint colours our KD suggested a neutral palette that would be carried through out the house. All of our furniture was in storage and so she based her choices on such things as our hardwood floor colour (mid-tone maple) our fireplace (squamish rock floor to ceiling), our front entry slate, and our granite. (GO in the kitchen and wenge in the bathrooms). So, look at the hard surfaces that you are installing and see if you can take inspiration from those things. After moving in we only wound up changing one colour and that was in the hallway.

The colours that she used were BM florentine plaster (kitchen, family room, LR and DR), BM stone hearth (rec room), BM ranchwood (MSBR), BM muslin (office) and BM white sand (bathrooms).

It then became a slow process of choosing things as we went along after moving back in. Some pieces were easier than others - for eg - coffee tables and area rugs were easy as I knew what I wanted but choosing new lamps was harder.
We have now been in our house for 18 months and have just finished selecting and ordering window treatments and we still need a new table for the eating area.

Take it slow and if possible, as others have suggested, live in the space for awhile. The thing that I discovered was that because everything was new my old tastes, while still workable, needed to be tweaked just a little bit to fit the new space more cohesively.

I'm in the boat that recommends you live in the house for awhile before doing any major decorating. We painted before we moved into our then new home and though much of the colors worked out fine I wound up changing many because the furnishings from our then current home didn't work in this home. Ofcourse the house styles were totally different. If you're new home is similar in feel to your last home then I'm with graywings on taking bits of your furnishings to the new home and seeing what you think will work wall color wise.
Good luck!