hey there,
I don't want to confuse you, but we just got back from Syria and Lebanon last week and had about the same amount of time as you for Syria and did HEAPS, so much in fact that we ended up spending a full week in Lebanon instead of the planned few days.
firstly, agreed about Lattakia, waste of time, boring place and while the train trip is quite nice near the Turkish border at the end, it's not worth the effort for the rest of it, very smokey and uncomfortable.
I would spend at least two nights each in Damascus and Aleppo and base yourselves at Hama for the dead cities etc, as it is much cheaper to do from there. We stayed at the Cairo in Hama. It's not the Ritz, but very friendly and GREAT value. They have a fantastic driver based at the Cairo hotel who will take you wherever you want for 700SYP p/p (which is incredibly cheap!). So save your money on the cheap hotel room which you'll only use for sleeping anyway while you do all the day trips and splurge in Damascus and Aleppo where you'll spend more time nearby.
this is what we did...
1) arrived Damascus late afternoon, 3 nights/two full days in Damascus for souqs and old city (could have easily cut this to two nights and spent an extra night in Palmyra)
2) left Damascus early took a bus to Palmrya, afternoon/overnight and day at ruins. fabulous.
3) bus to Hama, via Homs (another place you could base yourself for day trips) 3 nights in Homs. Depending on when you're there the waterwheels might be turning, they were just starting the day we left (midmarch). Homs itself is worth a couple of hours of wandering, very pleasant town. From there we did one day trip to the dead cities and Apamea, easily done in one day. Another day trip to Krak De Chevaliers, St George Monastery and another Crusader castle. There was the option of another day to the beehive houses, but we pushed onto Aleppo.
4) bus to Aleppo, two nights in Aleppo. We stayed in a hotel in the souq, Dar Halabia, which was my favourite hotel of the whole place, and cheaper and much more authentic than the very expensive "boutique" hotels in Damascus.
5) train to Lattakia, then onto Lebanon...
if you have extra time, maybe visit the Euphrates area...Deir-ez-Zor etc. or spend more time in Aleppo and surrounds, see some more dead cities or go to Beirut for a couple of days!
if you need more info, feel free to get in touch with me directly, it can be very confusing when you're there...
they didn't check vaccinations or anything really...
have a great trip, you'll love it!