Wow, my eyes do not see cherry as being dark red, at least not the cherry in my kitchen cabinets or my lumber stock.
I prefer not to stain or dye, but it is personal preference. If I had to change the colour of wood, I think I would use dye. Easier to control the colour and allows the grain to still show it beauty.
If you want dark, the non-stain/non-dye cheapest wood is walnut. I love the rich tones in some pieces. Can be cream in the sap, dark brown in the heart wood and a mix in between. I like working with walnut.
If I had to stain or dye I would likely use ash over oak.
Pricing is local, so what may be inexpensive for me could be more expensive for you.
If you have to go with softwood to save money, then consider poplar, although can be a challenge to stain / dye due to the natural colour variations.
You could consider contrasting colours, dark for the top and drawer front and doors, light for the frame.