why do you think your mobo is dying? gernally they either work or they don't. when the die they usually die suddenly, not slowly.
to answer your question, all modern AMD processors are good. it's not going to take much to outperform your 1.6GHz P4, any AMD processor above an AthlonXP 1600+ will perform better.
the Sempron is the budget AMD lineup. the Semprons were/are either rebaged older models or current models with limited/disabled features. even the slowest/oldest Sempron (rebaded AthlonXP3000+) is still twice as fast as your current processor.
the Athlon64 processor replaced the AthlonXP. it outperforms the AthlonXP and all Intel models prior to the Core2. AMD held the performance crown for years with the AthlonXP and the Athlon64 (that is, until Intel released the Core2 models)
Turion is not a processor, it is a package of components (including an Athlon64 CPU). it's just a marketting move to compete with Intel's Centrino line (which is also not a CPU, but a package of components including a CPU). there is no performance advantage to these packages.
also, you shouldn't write-off the Celerons so fast either. The Celeron is not it's own CPU model series, it is a generic name Intel uses for it's budget CPUs in certain models. each Celeron model is based off a higher priced model (Pentium4, pentiumM, Core, etc). to properly judge how well a Celeron will perform you need to find out which P4, PM, or Core model it comes from.
as an example, the laptop I'm using to type this on has a 900MHz CeleronM which in benchmarks performs as well as a 1.5GHz Pentium4. that is because it's based off a newer, more efficient arcitecture that gets more work out of each clock cycle.
in general, unless you get a really good deal, try to stay away from Pentium4s. they are inefficient, power hungry, hot running, and require high clock speeds to perform well. this is also true for the Celerons that are based off the Pentium4's.
if you get an Intel then go for the PentiumM's, Core's, and CeleronM's. these processors are much more efficient, consume less power, generate less heat, and don't need to run at high clock speeds to perform well.