As an old chain wears, the rollers get smaller and the chain settles deeper into the teeth of the gears. This results in more wear on the gears and, over time, the teeth will get pointier. Because the chain rollers are thicker on a new chain, it will sit higher on the teeth of the gears and when the teeth are worn enough the chain will slip.
Changing the length of the chain won't help. Contrary to popular belief, chains do not stretch significantly; the distance measured between the rollers is longer because the rollers become smaller as they wear. Past the .75 wear mark on a chain tool the gears will wear faster, at 1.0 the chain is at a higher risk of breaking.
The best solution for you at this time is to purchase and install a new cassette. Be certain to clean the chain and gears frequently to remove grit and metal shavings. Along with proper lubrication this should help keep the rear cassette in good condition longer.