The problem your describing is the syncro for 5th gear is worn out and giving you warning signs. The coasting in neutral, then putting it back into 5th gear, requires the 5th gear to be spinning at a speed to mesh without grinding. When you push on the gear shift, the resistance you feel is the synchromesh engaging. Every time it engages, it wears a little on it.
If the transmission oil is low, it will cause the syncro to wear out extremely fast. I would check or have the oil level checked. If you use to light of an oil, it will cause it to wear faster than not.
You can change the oil to a heavier weight oil, with a slight, probably not noticable difference in mileage and it will help some.
Although the transmission has to be pulled to have the syncromesh portion replaced, it is driveable as it is and if you don't mind, you can work around it, virtually indefinitely.
First, if you match the engine speed to the drive line speed, the gears will be already spinning at the correct speed to mesh properly. You can feel this point, occasionally, if you try to put the car in gear without depressing the clutch and the gear shift is able to move the gears to engage, without having the clutch released. The problem with this is that you have to match the drive speed and the rpm, for it to work. However, the same principle works even if you depress the clutch, if the rpm and drive speed are matched the gear can be engaged with little or no syncro work.
The other alternative is that if you shift into 4th, that uses the 4th gear syncro to spin the gears and get them to pretty close to the correct speed. If you then shift it into 5th, the 5th gear syncro doesn't have a lot of work left to do.
Early automobile manual transmissions did not have syncros. Instead, as the operator, you had to depress the clutch, shift into neutral, engage the clutch, speed up the engine to the right rpm, depress the clutch, shift into gear and then reengage the clutch. This technique required a certain amount of art/skill to estimate the correct rpm, to go with the speed being driven and the gear being selected. When it didn't get done correctly, you would end up grinding gears. When you didn't do it properly, very often, you'd blow out the transmission gears with chipped teeth and metal filings.
When you do the neutral coasting and gear reengagement, it puts a strain on the syncros. In my personal perspective, in and of itself, transmissions are far to big a repair job to fool around with saving a few cents in mileage, at the expense or potential expense of the transmission. Other people may feel differently.
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