The suspension is probably getting on a bit and a decent set of new shocks will do a lot to help with the handling. It would also be worth getting some poly bushes whilst you are at it. Which engine does yours have?
A lot of owners have gone the supercharger route for big power gains and it seems to be power that is the only thing lacking on these.
Hi Buddy, you can buy a set of coilovers very cheap on EBAY and they will transform the ride and handling, mine are Technics from Germany I paid £169 delivered.but you can get them even cheaper. Dont be put off by the price Loads of MX5 owners have them none have complaints Stay away from polybush use std bushes its cheaper and they suit the car better. you will then need the suspension geometry set up. You will think youve got a new car