Driving in 3rd gear instead of fourth will not affect your transmission as much as your engine. In fact, if you are driving in a manner that causes a lot of shifting between 3rd and 4th, it may be easier on the transmission to leave it in 3rd. If your transmission needs fixed, I don't think it would be from leaving it in 3rd gear. As far as cost, it will vary greatly depending on what needs done and what type of car you have.
More likely it has something to do with pressure problems in your transmission, check the fluid and replace the transmission filter. If the problem is not fixed check the vaccum lines running to the transmission and the vaccum off the engine.
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Assuming it is an automatic transmission, I would check the fluid level. When the transmission is warmed up, the level should read 'full' on the transmission dipstick. Check the color and smell of the oil. Does it smell burnt, or does it appear to be brown instead of red? Perhaps the filter screen is clogged.
Faulty transmission Pan gasket
transmission cooler
There are several things that can cause transmission overheating. One of the most common causes is overloading or overworking a system.
overheating of the fluid. This causes the transmission to slip. Slipping causes damage as friction material wears out quickly and even more overheating. This eventually causes failure.
sounds to me like it is too full. some trans. have to be checked in nuetral with engine running. it usually gives instructions on the dipstick
check your transmission fluid, if it is low it will slip
does the speedometer effect the transmission
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When was the last time you serviced the Transmission. Pull out the dipstick on the transmission while car is running with the transmission in park. and check the fulid level is it full of fluid?, if so smell the fluid does it stink or look dark? Then this may be all you need to do is take it to a Shop and let then service it for you.