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Immediately following core formation on Mars, the planet underwent a magma ocean phase. Volatiles released from the magma ocean fostered a primitive atmosphere which modulated heat loss from the cooling planet through the greenhouse effect. The solidification and degassing of the magma ocean are therefore coupled. In this work, we investigate two important aspects of this evolution: (a) the dynamics of melt trapping at the freezing front of the residual mantle and (b) the oxidation state during crystallization. For crystallization rates applicable to the martian magma, compaction is inefficient, leading to large fractions of melt trapped together with the crystals accumulating in the residual mantle. The H2O content of the martian residual mantle is strongly influenced by dynamic melt trapping. Following magma ocean crystallization, up to 55.4% of the initial H2O in the magma ocean is sequestered in the residual mantle, with the rest outgassed to the surface. Dynamic melt trapping also limits variations in trace element concentrations and fractionations. Resulting variations in important isotopic parent/daughter ratios (Sm/Nd, Lu/Hf) cannot account for all of the isotopic diversity inferred for martian basalt source regions, hence requiring alternative mechanisms. The redox state of the magma ocean exerts a strong control on the total CO2 content of the residual mantle and the time of crystallization. Under oxidizing conditions, the residual mantle stores 0.01% of the delivered CO2 but under the most reducing conditions we examined, the residual mantle can sequester 80.4% in the form of trapped carbonated melt and graphite/diamond. Plain Language Summary: The crystallization of the magma ocean phase of early Mars evolution is expected to affect the distribution of volatiles such as water and carbon. The amount of volatiles trapped in Mars' interior in its early evolution has been debated. It is generally assumed that most of these volatiles escaped the Martian interior early on to create a primitive atmosphere, leaving the interior nearly empty of volatiles. Water and carbon dioxide released during the crystallization of the magma ocean forms an early martian atmosphere that captures heat loss from the cooling planet through the greenhouse effect. In this numerical work, we show that this coupled evolution of the volatile degassing and magma ocean crystallization trapped much more melt and therefore more volatiles than previously thought due to the rapid freezing of the magma ocean. Our models show that up to 55.4% of the total planetary budget of water and up to 80.4% of CO2 as trapped carbonated melt and graphite/diamond can be stored in the mantle due to this previously unaccounted process on Mars. Key Points: Coupled evolution model of melt trapping and oxidation state of the martian magma oceanH2O and trace element content of residual mantle strongly influenced by dynamic melt trappingRedox state of the magma ocean controls CO2 content in the residual mantle [ABSTRACT FROM AUTHOR]