Do you know if coinomi handles the BCH and BSV coins well from old addresses?
Looking into this, I got warnings from Bittrex that I need to separate BCH and BSV coins before sending it to them or they could be lost somehow.
This is correct
You need to follow this guide to properly split your coins.
But remember, first of all: Move your Bitcoin to any wallet, with a different seed. Only then use coinomi and split your coins.
About replay protection:
There is no replay protection between the BCH (ABC) and the BSV chains. This means that, unless you properly "split" your coins, transactions you make with one of those coins will also be made on the other.
Sending unsplit BCH will result in the same amount of BSV being sent to the same address (and vice-versa). The receiving wallet may not be prepared to accept BSV into their address, so there is a possibility of loss of BSV coins. This is not an issue specific to Coinomi, as any other wallet you use is under the same risk.
How to split coins:
Splitting is only necessary if you had a positive BCH balance on November 15th 2018. If you received either BCH or BSV after that date, they are most likely already split.
Receive a new deposit into your BCH wallet of BCH that is already split. Any value will work, however small. The easiest way is to receive BCH from an exchange. If you are sweeping unsplit BCH/BSV from a paper wallet, you must still receive split BCH from somewhere else before moving to step 2.
After transaction from (1) is confirmed, open the BCH wallet and copy your current BCH "receive" address
On that same BCH wallet, go to the "send" section and send your full BCH balance to the address copied above.
Done. This should be enough for your coins to be split. This process only has to be done once. After your coins are split, they are split forever, and further deposits you receive will also most likely already be split.
https://coinomi.freshdesk.com/support/solutions/articles/29000026274-bch-abc-bsv-fork-information-splitting