As you've indicated, 137th Infantry was organic to 35th Infantry Division. Company A was an element of the regiment's 1st Battalion, one of the battalion's three rifle companies. Records are usually accurate except in cases where compiler(s) had to rely on guesswork. The regiment was operating in Luxembourg, at or near the Belgian border on 8 January 1945. "Hit the beach" - like "Jump off", etc. - was (and is) often an expression; one of many that aren't always to be taken literally.
If he's interred in an American cemetery overseas you should be able to locate him through the website at < http://www.americanwardead.com/searchww.htmhttp://www.americanwardead.com/searchww.htm > (click on the address to go there; follow directions when you arrive).
His deceased file should contain the date and location of his death and other information of interest. You'll find instructions on how to order at Message #2112.