Good work for getting out
I guess it's all relative. Better base layers will disperse moisture more effectively but of course how damp you get will depend on lots of factors besides this - it's definitely your enemy once you stop though; especially out in the cold and I've been blindsided by this a few times.
FWIW I have a fully synthetic RAB base layer and somewhat more expensive 80% Merino Finisterre item and haven't noticed an obvious difference in wicking, although I've not been specifically paying attention to this so that's not to say that there isn't a difference. The Merino does feel nicer on my manky, hyper-sensitive skin however.
I think the least-invasive option is probably to remove layers when you get to the cafe to let the sweat out - usually at the end of a ride I find I get warm for a bit due to the lack of windchill; especially if going straight into a warm building. Once you've let as much of the sweat flash off as possible, you can start putting layers back on again...
If you're really damp you could pack a second base layer / and get changed when you arrive to prevent the moisture making you colder.
I tend to do variations of this at work when it's less cold - arrive at the office, since I'm the only one there all top layers off, base layer draped over the bike to air.. when I start feeling the cold either I put some / all of it back on again or resort to the spare set of civvies I keep in the Brompton's front bag.
Ultimately temperature control under varying conditions is, IMO one of the more challenging aspects of cycling and I think really you'll only find the least-worst solution by experimenting..