I can't help you with companies, but I did buy a solid wood bedroom set which consisted of a deck bed with armoires on each side and mirror in the middle for $5,000., no chest or dresser. A salesman told me you will know the difference in solid wood furniture by the price. He was right!

Their are many different types of wood and they all have some what different characteristics, so I well designed and engineered piece of furniture will use a mixture of solid and engineered woods (plywood and mdf veneers). Also considering the type of finishes and the processes to apply those finishes. Manufacturers that have many years of experience in building high quality furniture is where I would start; Stanley, Century, Bernhardt to name a few. Kincaid is a option for solid wood furniture.

If you want solid wood furniture I would look towards the smaller niche furniture makers and stores, i.e. Pompanoosuc Mills, Copeland Furniture, Chilton Furniture.

You should go for online research. Many good online stores are available with high quality furniture with affordable price-list.

I'm in a similar situation. Our furniture was all ruined by a water pipe flood while we were away. The house, which had to be gutted, is repaired (took a year), and the GC has been paid (took another year). Now we need to replace all the furniture. I looked into the companies suggested by sloyd. Thanks sloyd!
Chilton has several pieces that are simply made, and, while not as stylish as the ones I love at Room and Board, are much more affordable. I'm also looking into Gat Creek, which I can buy through Hickory Furniture Mart. We like the look of solid, natural cherry pieces with clean lines. What are you looking for? I'll be very interested in any good sources that are suggested here. Please let me know how you are progressing. I have some idea what you have been through. Good hunting.