First off, yes both MD5 and SHA1 have been retired from cryptographic use because because it's possible to construct pairs of messages which produce the same hash value (aka "collisions"). That said, the chance of hitting a collision at random is absurdly low.
MD5 has a 128-bit output, SHA1 is 160 bits. Let's assume that a file being hashed will hash to one of the 2128 / 2160 possible outputs with equal probability, then on average, you expect to find a collision after hashing 2127 / 2159 different files.
Say you filled a hard drive with 512-bit (64 byte) files. You would need 2133 bits = 1.3 x 1027 TB for MD5, or 2165 bits = 5.8 x 1036 TB to store these files. For comparison, it's estimated that by 2020 the human species will be producing 44 zetabytes (4.4×1010 TB) of data per year.
So even if you hashed every file on the planet with MD5, you only have a 1 in 1026 chance of getting a collision by random chance.
To answer your original question:
To remedy this I thought I would switch to SHA1, but then had the thought that if I compared both, there would be even less chance of collision.
Does anyone here know both algorithms well enough to confirm or deny this?
Yes, it's true that since they are different hash functions, the chances of there existing two messages which are collisions for both of them is minuscule. Whether it's less minuscule than the chance of collision in SHA-2/256 or SHA-2/512, I don't know. At this point we're debating astronomic probabilities (ie the chance of getting a collision is smaller than the number of atoms in the universe type thing) so /shrug ?
Now, if you are talking about malicious attacks (ie files specifically crafted to fool MD5 or SHA1), then maybe you have a point. ... In that case use SHA2 (which is 256 bits) and you're fine :)