Is this secure?

Yes, One-Time-Pads (OTPs) can be proven information theoretically secure. For a sketch of what this means and how to do this, please refer to this previous answer by me.

Can I actually use modular addition as encryption like it said in Wikipedia?

Yes, any group operation can be used to form a pefectly secret encryption scheme similar to the one time pad, given that the operands are both part of said group, this includes, but is not limited to:

• Addition over the real numbers $\mathbb R$ (not recommended due to potencial issues with CPUs rounding floats)
• Multiplication over the complex numbers $\mathbb C\setminus \{0\}$ (not recommended due to potencial issues with CPUs rounding floats)
• Addition in $\mathbb F_2$ (also known as "XOR" or one time pad)
• Addition in $E_{a,b}(\mathbb F_p)$ (the set of points on the elliptic weierstrass curve formed over $\mathbb F_p$)
• Addition in $\mathbb F_p$ (also known as modular addition)

This was also acknowledged in Introduction to Modern Cryptography by Katz-Lindell (2nd edition) in Lemma 11.15 and right below (while discussing ElGamal encryption). The idea of the proof there is that you use the fact that the key is uniformly taken from the set of the possible keys and that you "transfer" this randomness into the ciphertext.

And the plaintext is as long as the OTP, so when I send (plaintext,OTP), do I first (plaintext+OldOTP) and (NewOTP+OldOTP) and then combine them, or do I do something else?

1. Normally, you wouldn't actually send the OTP along with your message (in clear), because in this case the OTP is of no use.
2. Note that you use "OldOTP" twice, which contradicts it's name one-time-pad (OTP) and is thereby insecure.
3. Note further that you can't use the OTP to transmit a longer OTP because using an $n$-bit OTP you can only transfer an $n$-bit message and thus only an $n$-bit OTP which didn't help you at all.
4. The usual usage of the OTP would be to transfer the OTP securely to the recipient and then using it only once with a message, as in (Message + TheOneAndOnlyOTP)

Is this secure?

Yes, One-Time-Pads (OTPs) can be proven information theoretically secure. For a sketch of what this means and how to do this, please refer to this previous answer by me.

Can I actually use modular addition as encryption like it said in Wikipedia?

Yes, any group operation can be used to form a pefectly secret encryption scheme similar to the one time pad, given that the operands are both part of said group, this includes, but is not limited to:

• Addition in $\mathbb F_2$ (also known as "XOR" or one time pad)
• Addition in $E_{a,b}(\mathbb F_p)$ (the set of points on the elliptic weierstrass curve formed over $\mathbb F_p$)
• Addition in $\mathbb F_p$ (also known as modular addition)

This was also acknowledged in Introduction to Modern Cryptography by Katz-Lindell (2nd edition) in Lemma 11.15 and right below (while discussing ElGamal encryption). The idea of the proof there is that you use the fact that the key is uniformly taken from the set of the possible keys and that you "transfer" this randomness into the ciphertext.

And the plaintext is as long as the OTP, so when I send (plaintext,OTP), do I first (plaintext+OldOTP) and (NewOTP+OldOTP) and then combine them, or do I do something else?

1. Normally, you wouldn't actually send the OTP along with your message (in clear), because in this case the OTP is of no use.
2. Note that you use "OldOTP" twice, which contradicts it's name one-time-pad (OTP) and is thereby insecure.
3. Note further that you can't use the OTP to transmit a longer OTP because using an $n$-bit OTP you can only transfer an $n$-bit message and thus only an $n$-bit OTP which didn't help you at all.
4. The usual usage of the OTP would be to transfer the OTP securely to the recipient and then using it only once with a message, as in (Message + TheOneAndOnlyOTP)