Does indeed sound like the usual worn out transmission.
As said above perform a transmission service (Drop transmission pan, replace filter, refill fluid). Check the condition of the oil currently in the transmission, if the oil smells 'burnt' and is dark brown this is usually a sign of worn/burnt clutches but a service may help keep it alive longer. If the service makes a noticable improvement would strongly recommend fitting a trans cooler as the lower operating temperature will certainly help keep the trans alive longer. A service and cooler is definitely cheaper than a rebuild. Keep in mind that high RPM shifting into 2nd and 3rd will put a lot of strain on a worn band or worn clutch packs.
Most likely culprit is a worn 2-4 band or burnt 3-4 clutch plates. 3-4 Clutch plates are prone to failure in these transmissions and can be replaced with a 'powerpack' to increase durability (part of a rebuild). The 2-4 band can be adjusted with the use of shims on the band apply pin in the band apply servo assembly but would only recommend doing this if you were confident in modifying your transmission. The seals and o-rings in the servo assembly will also need to be replaced if doing this as the whole assembly will need to be removed from the transmission to install shims. This will need to be done while the pan is removed from the transmission so the band is visible to check clearance. The installation of shims is also a part of the installation of a corvette servo which comes with the necessary seals and o-rings for the assembly. Installing a corvette servo isn't recommended in a worn transmission though as it will increase apply pressure which will exacerbate any problems with worn components.