Only amphetamine and morphine based drugs need a prescription. Most stuff is available over the counter and the prices are cheap. Most GPs will give you a prescription for 3 months supply if you tell them why. In the very unlikely event of you getting stopped on entry make sure you have a copy of your prescription.
I take it that you have a re-entry permit? I can't see any reason to transfer it as you're not renewing. The only problem is the IO on exit may make a fuss.
The 60-day entry is not a visa it is a visa-exempt entry. As you are returning so quickly, I suggest you book an onward flight within the initial 60-day free entry, a hotel booking for the first night and carry 20,000 baht in cash. This is in case you are selected for secondary screening due to repeated entries. You should get a Destination Thailand Visa, which gives you 180-day entry and lasts 5 years. You can book an onward flight for about $14 - there are many websites but the ticket is only valid for 48 hours so book just before you travel.