One of the reasons you were left with marks on the bowl wall, is because scraping is usually one of the last things done. This can cause problems because the walls of the bowl are thinner, (at the end of turning) and more prone to vibration. The scraper puts a fair amount of stress on the timber and if the walls are thin, it will cause chatter, uneven gouges and marks like you described.
Thicker walled bowls seldom cause problems when scraping the inner walls. However, the trend in turning these days are bowls that are thin enough to see through (just kidding). :-) At your next club meeting, get one of the advanced turners to show you how to shear scrape. When you become proficient at it, it will make your life much easier and reduce sanding considerably.
Having said that, if you have a good technique with your bowl gouge, you should rarely need to use your scraper at all. Some of the exotic timbers actually scrape better than they cut, but that is another kettle of fish.
If your bowl gouge has a laid back Irish grind, you may have entry problems, whilst working on the upper wall. This is caused from the steep angle of the gouge. You may want to consider getting a secondary gouge and grinding it with a traditional straight across grind. Straight across bowl gouge grinds permit a faster swing and allow you to maintain bevel contact on radical sweeps inside the bowl.
Have you tried the step-technique to work the inside of the bowl? When using the step-technique, the inner bowl is completed in a series of steps. This reduces the potential of vibration and chatter considerably. It also allows more freedom in the turning tools used, because of the extra support given by the unturned inner mass.
For example, after you have trued the exterior of the bowl, work your rim. After your rim is completed, begin to form the inner curve, about one to two inches at a time. The unturned mass left on the inner walls of the bowl, helps to reduce any possible chatter or vibration, whilst you reduce the inner wall. When you have completed the first two inches, extend the inner curve, two more inches. Do this until you have completed the inner curve.
If you use this method, you will never have to go back up the wall to smooth anything out. While you may see many seasoned or professional turners taking the inner cuts in one sweep, it takes practice to achieve this level of skill. Especially when the walls get thin and want to chatter on you. When you progress further in your turning, you will be able to cut with the best of them.
On the other hand, a shear scraper can often be used up the wall with no adverse reactions, perhaps because the shear scraper is actually cutting rather than scraping the surface and is applying less outward pressure.
Yes, the scraper can work very well in smoothing up the inside bottom of a bowl where there is still a lot of stability to the wood and the rotational forces are less. one hint passed on to me by a professional turner: before hollowing-out the last two-thirds of the bowl, get the rim and first third to the thickness and finish of the final product. Then complete the last two-thirds, working down the side-walls as completed in the first third.
If you keep in mind that as you hollow the bowl, the more out-of-round the sides become, scraping can be limited to the bottom.
The tool use like this is a lot less likely to grab and cause vibration and with practice (that means after you've destroyed plenty of bowls) it's fairly easy to get a light whiskery touch over the surface. The scraper does need to be kept ground for this to work best. You'll find arguments for and against leaving the burr on the edge, but none from me.
One bit of advice I pass on, it's not my own but something that's rattling around in my head from somebody else. If you're making thin walled bowls say 1/8" or thereabouts, then make a 1/8" bowl right from the start, don't make a 3/8" bowl then thin it to a 1/4" bowl and then think I can make that a bit thinner. It'll have probably moved a bit in the making and you won't get uniform wall thickness. Plus if you make it 1/8" at the top to start with and you still have bulk in the rest of the bowl you'll get cleaner cuts.
I assume that the tool rest is at a height that the cutting edge would be at center with the tool horizontal. Then raise (rotate) the handle up to the 2:00 position so that the edge is below the center and at the 8:00 position. The shear scrape is then below the center.
There are three basic approaches;
As far as height above the centre line goes the general rule of thumb is to scrape above centre height. This is because the diameter of the vessel at this point is smaller than its maximum diameter at centre height. Any tendency for the scraper to catch will throw the tip of the scraper downwards. If you're working above centre this downward motion should throw the tip into an area where the diameter of the vessel increases and by so doing place the tool into unoccupied space. Working below centre, the tip will be thrown down into an increasingly smaller diameter and will slam the tool into the side wall of the vessel.
Having said that the shear scraping action lends itself less likely to promote big catches. With shear scraping, you need a good surface to start with as you are going to follow any slight undulations that may be present.
Have the tool pointing slightly downwards as you shear scrape, in other words raise the handle half an inch to an inch and try to glide the tool across the surface of the vessel, you're only trying to remove whiskers of material at this stage.