All the Nav Logs I know give MSAs far higher than 10nm either side.
Your ops. manual should have a note about how the nav. log is constructed. The two I have used both say that the MSA on the log is calculated by inspecting an area 10nm either side of the planned track and taking the highest grid safety altitude (from an AERAD or other nav. chart) that falls into that area for each route segment. Obviously one would hope that any high stuff around would be in the pilots mind anyway and they don't go too far off track but a much higher MSA than you have printed on the log could in theory be very close.
The ICAO document seems fairly plain but I still wonder in busy areas if you a creating as many problems as you solve by turning, there is traffic everywhere anyway and a turn is certainly not going to put you clear of traffic. As for TCAS, if you have the presence of mind (and I not sure I would) you should set it to TA only (as for an engine failure) so that it does not give you an RA and any unit in a conflicting aircraft will know not to try and co-ordinate with yours. Would be a pretty interesting RA for the other guy though, two emergency descents in formation, lovely!.