To reduce the MW I think the esiest way is by mechanical work, either with high speed agitation of highly concentrated solution of lignin or by a fast passing through a constrained geometry (ceramic sieve).

Hydrophobically modified polymers and polyelectrolytes is among the most investigated topics in polymer science. Grafting an alkyl chain (radical) into a lignin backbone is not a hard task regarding the various reactive sites in lignin chains, and there are a lot of examples in lterature.

Comaring to redu cing hydrophilicity of sodium lignin, I think it would be better to directly use other types of lignin (not lignosulfonate), they will show much better hydrophobicity.

You might alkylate your kraft lignin preparation and thus reduce its hydrophylicity.

It is possible to perform with diazomethane /you obtain methylated lignin/. You have to be careful and follow the corresponding safety standards. CH2N2 is an explosive gas sensitive to friction and heat! From this reason, diazomethane method is suitable only for methylation of very small samples.

Another way of hydrophobiation is alkylation /performed in water suspension of lignin/ with dialkylsulphate /dimethylsulphate, diethylsulphate..etc../ in the presence of NaOH.

Alkylation will convert phenolic and aliphatic hydroxyl groups to alkyl- and aryl-ethers  and make kraft lignin less hydrophylic.

Both methods of alkylation, in general, are described in any preparative organic chemistry book.

Kraft lignin /or any other/ can be hydrofobized also via acylation /substitution of H in  alcoholic and phenolic OH groups of lignin with acyl- /.

You may disperse the lignin preparation in 99% acetic acid, then add acetanhydride to achieve 50 % V/V concentration, and heat the mixture at 80 oC  for 30 to 60 min. Lignin to acetylation mixture ratio might be 1 to 4 or higher. Perform acylation in a closed flask and in digestorium. Then pour carefully the solution of acetylated lignin /dropwise/ into ice-cold distilled water /vol. several times higher than that of acet. mixture/.  Neutralize dilluted acetic acid with solid Na2CO3. Acetyl derivative should precipitate slowly. Centrifuge or filter off the product, wash with water and carefuly dry it.

Acylated lignin can be separated also via rotavap at 80 oC, the dillution of spent reaction mixture and neutralization of excessive acid should be omitted.

Instead of acetanhydride you may use propionic acid anhydride in propionioc acid. The product, in this case, should be more hydrophobic due to longer chain of aliphatic carbons CH3CH2- /2/  

Note: instead of anhydrides the corresponding acyl halogenides /R-CO-CL or R-CO-Br can be used. The procedure of acylation is simlar to that above mentioned, however it is faster and higher attention is requested to dillution of the final reaction mixture with water and to neutralization. Particulars can be found in any preparative organic chemistry book. Chemical stability of the prepared esters is somewhat lower /hydrolysis/ as compared to lignin ether derivatives.