As Smack has said the work can be quite varied and your options after uni are not limited, as I've got a MechEng degree but work in finance now but have also worked for engineering companies. My uni did both hand drawings and CAD drawings. You're right there is a lot of maths, only few modules that won't have any maths content like design sessions, interpersonal skill sessions. Computer work (I assume you mean CAD and similar software) is becoming more frequent, as you can produce engineering drawings using software (it's a lot easier and quicker). Especially in final year you will be using CAD and few other programs like Matlab or Ansys as you will have some sort of design project.
Is it satisfying? Depends on if you like solving problems using maths. The idea of the frst two years is to give you a good grounding in maths, dynamics, fluid mechanics etc so that you use those for real-life problems which most likely will be part of your dissertation.
Lucrative...again depends, oil & gas sector pay well and plenty of opportunties abroad. But again it depends on what sector you want to get into.
Me personally, I loved it. I love maths, physics and generally anything that is challenging. And thats what engineering is, its challenging. Work load is quite heavy though, so you won't have as much free time as people on other courses and especially in final year, it's a lot of work. If you like cars/racing then there is chance to be involved in formula student too.