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Very recently, the LHCb Collaboration reported the doubly charmed tetraquarkstate $T_{cc}^+$ below the $D^{*+}D^0$ threshold about $273$ keV. As a verynear-threshold state, its long-distance structure is very important. In themolecular scheme, we relate the coupling constants of $T_{cc}^+$ with$D^{*0}D^+$ and $D^{*+}D^0$ to its binding energy and mixing angle of twocomponents with a coupled-channel effective field theory. With the couplingconstants, we investigate the kinetically allowed strong decays $T_{cc}^+\toD^0D^0\pi^+$, $T_{cc}^+\to D^+D^0\pi^0$ and radiative decays $D^+D^0 \gamma$.Our results show that the decay width of $T_{cc}^+\to D^0D^0\pi^+$ is thelargest one, which is just the experimental observation channel. Ourtheoretical total strong and radiative widths are in favor of the $T_{cc}^+$ asa $|D^{*+}D^0\rangle$ dominated bound state. The total strong and radiativewidth in the single channel limit and isospin singlet limit are given as$59.7^{+4.6}_{-4.4} \text{ keV}$ and $46.7^{+2.7}_{-2.9} \text{ keV}$,respectively. Our calculation is cutoff-independent and without prior isospinassignment. The absolute partial widths and ratios of the different decaychannels can be used to test the structure of $T_{cc}^+$ state when the updatedexperimental results are available.

Probing the long-range structure of the $T_{cc}^+$ with the strong and electromagnetic decays