One of the more tried and proven techniques for determining whether or not you can successfully hand hold your camera and lens combination is to compare the reciprocal of your effective focal length with your shutter speed.
For instance, if your effective focal length is 250mm, you should seriously consider using a tripod or some other means of support for shutter speeds less than 1/250 sec.
Using the information you provided (270 mm with a shutter speed of 1/125), you should be using a shutter speed of not less than 1/250, and preferably higher ... 1/500 or more; the possibility of getting a shake free exposure using 1/125 sec is questionable unless you use other techniques to reduce or minimize the effects of camera/lens movement during exposure.
Since you are already up to an ISO of 400, your options come down to:
Using a tripod;
Increasing your ISO to get more speed which will result in more noise/grain;
Flash (maybe out of the question considering distance involved);
The support you use doesn't necessarily need to be a tripod. I routinely shoot wildlife photos with a 600 mm lens using a sturdy monopod. Monopods are extremely versatile; they are less cumbersome than a tripods, can be quickly set in place, and provide you with considerable flexibility.
Another point to consider ... does your telephoto (Tamron) have a built-in tripod collar?
If you try to mount longer camera/lens combinations using the camera tripod mounting screw, what results is an extremely front-heavy camera that is almost impossible to hold steady even with a tripod - the camera/lens combination needs to be supported somewhere near the center of balance for that particular camera/lens combination - that's where the tripod ring on the lens comes into play.
Hope this helps ...
Ed
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