I did a West East crossing in 2002. Route was Morocco, Mauritania, Mali, Algeria, Tunisia, Libya Egypt. As Chris mentions ,you have to come up to the coast to cross into Libya and Egypt so we headed south again in both those countries having crossed the border to see the Sahara proper in those countries. Most traffic crosses the desert North South so teaming up with people is difficult.
As Richard mentions you'd be advised to take a guide for some parts (Mali (Gao)- Algeria (Tam.)) and I gather from neccesity in Algeria and Egypt now. Guides are not cheap but certainly give you peace of mind and can be great.
For my money I'd make my object Timboctou - plenty of desert on the way down and a fitting objective.
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