If you get an early start and begin the walk to the beaches before 9am, there's a good chance that you'll see monkeys. Last time I went to Tayrona, we saw several troupes of Howler and Capuchin monkeys while walking to the beaches. I have seen the Cotton Top Tamarin before, but you probably won't see it on the trek to the beaches from the park entrance. I stumbled upon them when I went into the forest, well away from the beaches. They weren't too happy to see me, and I got the impression that they aren't too keen on being around people.
If you're looking for the "jungle feeling", then Tayrona might not be the best destination. The area of the park by the beaches is pretty crowded and the forest is dry. Sure, you can walk a few minutes on some random trail and get away from the crowds near the beaches, but the infrastructure in the park isn't great and the trails aren't marked or taken care of. Also, the park's popular areas are very crowded and the people running the park don't seem to have too much of an interest in improving the situation. Tayrona is a patchwork of private landowners and the Colombian government has no stake in its day to day operations. The entry fee you pay to enter the park, isn't used on the park itself. Also, there are an odd number of people working and living in the park, who don't care much about preserving Tayrona's natural space. They often run noisy generators, burn things, and the trails are littered with the excrement of their animals.
If you're looking for an alternative to Leticia and the Choco, then I would recommend Mocoa in the Putumayo region. It's a bit out of the way, but it can be easily reached by bus. There's plenty of animals to see and things to do in the area. I saw several species of Monkeys while I was there and got a very close look at a flock of Toucans from the premises of my hostel.
I'm not sure when you're coming to Colombia, but these guys are starting a route to Leticia in the next few months:
https://www.vivacolombia.co
Their fares are often not much more than the price of a bus ticket, and they are a good option for those traveling on a budget.