1). In industry, methylamines are removed by extractive distillation.
https://www.researchgate.net/publication/245329246_Extractive_Distillation_A_ReviewExtractive distillation can also be effectuated in the lab, but a special glassware and adequate experience are necessary.
2). Alternatively, you can remove MeNH2 by transferring into HCl solution, as you did, but you have to cool your trap because the solubility of gases (HCl and MeNH2) decreases by the heat that is generated during the exothermic neutralization.
3). A more risky way that might work is adding dimethylamine, which forms a ternary azeotrope with MeNH2 and water, at about 30-40oC and then azeotropy with ethanol at 78oC.
4). Methylamine has a relatively high latent heat of vaporization (200 cal/g = 837 J/gr) and therefore, evaporation is slow. Thus, you need enough time, in order to remove it under vacuum, after alkalinization.