You can get a quality camera body from any vendor (it isn't necessary to switch systems). You'll find you can do quite a bit with the camera you already have and the best improvements will likely come from lens selection (and learning to process the data effectively).
Serious astrophotographers will often use a modified camera. Much of the deep-sky emission nebulae tend to glow in Hydrogen alpha (Hα) light. This is inside the visible spectrum but it's near the long-end of the spectrum and human eyes are not particularly sensitive to it. Photographers want photos that resemble what their eyes saw. Since human eyes are not particularly sensitive to these reds, they filter the camera sensor to block a considerably amount of that red light. At the Hα line (656nm), a traditional camera is filtering out about 75-80% of that light. A modified camera removes this filter and replaces it either with clear glass, or with a filter that doesn't block any visible spectrum light but does a hard cut-off at the near IR point at 700nm. This helps the camera collect 4-5x more light in those deep-sky emission nebulae. The modified camera is usually a used DSLR purchased at a bargain price (but still working) and then taken apart and modified. There are commercially produced cameras designed for astrophotography but most of these are not DSLRs (they are meant to be controlled via a computer ... and usually using a telescope.) If you were really wanting to invest in a new camera dedicated to astrophotography... that's what you would want.
There are quite a few different types of astrophotography. I tend to do long exposure imaging of specific deep-sky objects with camera attached to a tracking mount (and often through a telescope).
Often when I see this type of question, the type of photography being referenced involves using a stationary (non-tracking) tripod to shoot an image of a scene with a sky full of stars above (often using the Milky Way above a landscape).
If this is the type of astrophotography you want, then the important piece of gear is the lens ... not so much the camera body (although there can be advantages to some camera bodies).
The Milky Way is not particularly bright. Images you have probably seen featuring the Milky Way are enhanced to improve the contrast (the histogram is "stretched" to bring out more detail). Learning how to post-process the image data is a big part of this type of photography.
The main challenge is that the Earth is rotating from West to East at about 15 arc-seconds per second. This means that a long-exposure image may not have pin-point stars, but may instead have elongated stars due to the rotation of Earth.
The formulas for determine how long you can expose without noticing elongation of stars always involve knowing the focal length of the lens as well as the size of the camera sensor.
The simplest formula is the 500 rule.
500 Rule
This rule was created for cameras using 35mm film. A single frame would measure 36mm x 24mm (the name "35mm" refers to the entire width of the film ... including the sprocket holes used to advance the film inside the camera ... not just the exposed area.) A digital camera with a full frame sensor would also measure 36mm x 24mm.
The rule says that you can divide 500 by the focal length of the lens to arrive at the number of seconds you can expose without noticing elongation of stars.
If you have an APS-C sensor (such as your T3) then you have to divide 500 by the crop-factor (1.6 for Canon APS-C, 1.5 for Nikon & Sony APS-C). 500 ÷ 1.6 = 312.5
Assume using a typical 18-55mm f/3.5-5.6 standard zoom (the lens that is commonly included with a camera + lens kit) at the 55mm end. This would work out to 312.5 ÷ 55 = 5.7 seconds (not very long). Also the lens is limited to f/5.6 as the widest possible aperture at the 55mm end.
On the other hand, if you use the 18mm end, then it works out to 312.5 ÷ 18 = 17.4 seconds (3x longer) and you also get to use f/3.5 (1.3 stops brighter or 266% more light). 3x longer with 1.3x brighter = nearly 8x more total light (much better).
You can see where this is going... shorter focal length lenses with lower focal ratios give you an advantage when shooting night-sky scenes such as Milky Way shots. If you have money to invest ... this is where you'd want to invest.
Sigma makes a 14mm f/1.8 lens. It's about $1600 so I realize this is probably out of the question given your being a student with not a lot of money to spare. But I mention it because this would collect roughly 4x more light than an f/3.5 lens ... and you can stretch the exposure to 22 seconds (about 20% longer exposure).
Perhaps more realistic for budget is the Rokinon lenses which are completely manual. Those lenses are a few hundred dollars. They are completely manual (even the aperture is controlled via a manual aperture ring on the lens barrel). Some of these are fairly popular among Milky Way photographers. I do not own one but have read that it isn't uncommon to encounter a copy with 'de-centered' optics. This would mean that you'd don't get symmetric focus across the field... one side might be focused and the other side might be soft. If you choose to go with such a lens, test the focus symmetry carefully to make sure you don't have a bad copy while you are still able to exchange it without a hassle.
I should mention (for completeness) that there are other formulas for coming up with exposure duration such as the NPF formula. The 500 rule isn't the only rule.
Tracking Heads
I did mention tracking mounts earlier. These remove the time-constraints and allow you to shoot much longer exposures.
The idea is that as the Earth spins from West-to-East on it's axis, the tracking mount is angled so that it's axis of rotation is parallel to Earth's axis of rotation. The mount rotates from East-to-West at the same speed. This effectively cancels the movement so that stars remain stationary in your camera frame and you can take much longer exposures and still have pin-point sharp stars.
The two major vendors for these are iOptron and Sky Watcher. Both companies actually make two tracking heads. They each have a higher-end tracking head that is a bit beefier and can handle more weight than their lower-end tracking head. Prices range from about $300 to about $400 dollars depending which version you select and which accessories you include.
Another advantage of these heads is that you can use much longer focal length lenses to get longer detail shots of deep-sky regions.
Sky Watcher makes the Star Adventurer head and the Star Adventurer Mini head.
iOptron makes the Sky Tracker head and the Sky Guider Pro head.
