You don't need a cyclocomputer with cadence sensor to figure out your cadence but having one makes life easier. With a cyclocomputer you can check it at a glance to see if you're not keeping it up where it should be. When you're counting you're focused on the cadence and may be subconsciously speeding it up.

If you're used to 50 to 60 then 80 to 90 might seem really fast and really tiring. Don't get discouraged if it takes awhile before that feels comfortable. Note: Although 80 to 90 is often recommended, for some people that may be unnecessarily high.

If speed is important, learn to use the downhills to your advantage. It's free speed, don't just coast. I don't know what kind of bike you have but aerodynamics play a bigger role the faster you go, so getting down lower can help.