Heathcote, 21, arrives from Wythenshawe Town and previously made 51 appearances for Altrincham, as well as turning out for Hyde, Trafford and Ramsbottom.
Former Tranmere defender Shaw arrives from Marine, where he made 46 appearances last season.
Widnes co-manager Kev Towey said: "Danny has a huge amount of experience having played in both the Welsh Premier League and the Northern Premier League over the past four years.
"He is an exceptional talent with the correct attitude.
"Sam had fallen out of love with the game and dropped down a couple of leagues to start enjoying his football again.
"But he's now very hungry to be a success at this level, and he'll be a very important player for us at both ends of the pitch."
They are the third and fourth signings of the off-season, following Josh Klein-Davies and Lewis Buckley.
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