The recycling of waste tires avoids the environmental hazards of landfills and incineration, and its application in asphalt modification achieves resource sustainability. Currently, desulfurized rubber powder (DRP) is widely used as an asphalt modifier, mainly mixed with SBS, and fewer studies have been conducted on high-dose asphalt modification with ethylene vinyl acetate (EVA). In this paper, DR/EVA-composite-modified asphalt (DR/EVACMA) was prepared using 20% DRP and 4% EVA by adding four additives: furfural extract oil (FEO), a crosslinking agent (DCP), a vulcanizing agent (sulfur), and a silane coupling agent (KH-550). The aim was to study the effects of different additives on the physical properties, storage stability, and rheological properties of asphalt. First, conventional physical property measurements were carried out, and the data were analyzed using a polar analysis to determine the degree of influence of the four additives and the optimal ratios. Then, the rheological properties and fatigue resistance of DR/EVACMA were investigated through temperature scanning experiments, linear amplitude scanning (LAS) experiments, and multi-stress creep (MSCR) experiments. Finally, the reaction mechanism and microscopic properties were analyzed through infrared spectroscopy experiments (FTIR) and fluorescence microscopy (FM). The results showed that FEO had the greatest effect on asphalt characteristics. Compared to matrix asphalt and additive-free asphalt, DR/EVACMA has higher physical properties, fatigue resistance, and high temperature rheological properties due to its internal crosslinking structure. Its storage stability is also very good, with a difference of only 0.7 °C in the softening point.