No, dimethyl amine and triethyl amine are not more soluble in water. Both are soluble in water but not more soluble in water. These are also soluble in alcohol.
Dimethyl amine is a secondary amine while triethyl amine is a tertiary amine. Generally, the small amines are more soluble in water. As the hydrocarbon chain goes on increasing, the solubility goes on decreasing.