As for "new technology" -- excuse me? Clothes are dry when they are dry. Nothing new about that! The toilet analogy was apt: use less water, but requires several "go's" for it to "go"!
I don't believe that is entirely correct.
I don't have the time to pull up the references but there are some neat charts on appliances on the likes of the Siemens website, and also US National Research Labs have done the homework. See also Energy Information Administration. The dramatic move is in fridges (down 60-75% in 30 years on total pa consumption of electricity) but wet appliances have also made significant improvements.
DC motors are a lot better than they were-- that saves juice-- variable speed controllers etc.
Spins are a lot faster at least in washing machines (not sure dryers) -- that saves a lot on the heating element.
The modern ones have moisture and weight sensors so they adjust to how big the load is and how wet-- don't overdry. That's a big saving.
There's the whole point about condenser dryers (marginally less efficient) v vented ones. So technology change there, too, if not improvements in drying efficiency per se.
And then we get to 'heat pump' dryer technology: Siemens, Bosch, Miele all do them now. (Siemens and Bosch are the same company). The energy use drops are dramatic (c. -60% vs. a modern 'conventional' dryer). At my electricity price (USD 22 cents) that gives me a sub 5 year payback. If you have time of day pricing (we will) then the payback could be much quicker than that.
I'll stick with my original advice which is OP should not replace if at all feasible. Because there is an environmental cost in building a new machine. And all things being equal they are more complex and therefore likely to be less reliable.
Exceptions would be in water short areas (go for a front loading washing machine) and fridges/ freezers, where if it is pre 1992 the savings are extraordinary. Even at US electricity rates (c. 10.5 cents/ kwhr but over 30 in California) this can save $10-20 pcm. Also for a fridge you ensure safe disposal of the CFC Freon, one of the worst environmental catastrophes of the 20th Century-- you don't want that to leak out into the atmosphere. There are incentives from the utility if you are in a water short area (my parents received one in Ontario).