Either solution or gas should be much the same, solution is easier to handle so I'd try that first. Keep the amine in large excess to minimise tetra-alkylammonium salt formation. How hindered is the alcohol? By the way, if it is primary (or achiral secondary) it might be more effective to oxidise to the aldehyde and introduce NMe2 by reductive amination. Altenatively you could displace the mesylate with azide and then reduce the azide with H2/Pd with excess formaldehyde.
R-OMs + HNMe2 -----> R-NMe2 (possible side reaction: R2NMe2+ MsO-)
Alternatives:
R-OH ----> R=O --HNMe2, ([H2, Pd/C] or NaBH3CN) --> R-NMe2
R-OMs --NaN3--> R-N3 ---H2CO, H2, Pd/C ---> R-NMe2