「つつ」in the sample sentence is used as conjunction, linking two events and explaining those events are happening at the same time. This つつ is always pronounced with the immediate previous word; 「泣きつつ」should be pronounced as if they are a one word. If you separate 泣き and つつ, it sounds a bit unnatural. 「つつ」itself is pronounced just as is, tsu tsu, 2 syrable.

There is another way to use 「つつ」, which describes changing its stage/situation and so on. i.e., 寒くなりつつある(it's getting cold) 良くなりつつある(it's getting better)

Sample recording:
人は泣きつつ生まれ、泣き言を言いつつ暮らし、失望落胆して死ぬ
確認しつつ進める
考えつつ歩く

寒くなりつつある
良くなりつつある
増加しつつある

五つ
包む
うつつ
慎む

突付く
in casual conversation, you can also say つっつく

Hope it helps ;)