There are plenty of languages that do what you are looking for. In linguistic typology, languages that encode grammatical functions (such as tense) as separate words are called "isolating" (one mnemonic that I use is "they isolate the different meanings into different words"). Those separate words are frequently referred to as "particles."
Polynesian languages tend to have isolating verbs with a lot of tense particles. Some examples from the Wikipedia page on Hawai'ian (https://en.wikipedia.org/wiki/Hawaiian_grammar#Tense,_aspect,_and_mood)
ua + verb: perfective aspect, past tense; or perfect tense/aspect (ua hana au "I worked", "I have worked"). Note that the pre-verbal marker ua is often omitted in speech.
i + verb: past tense (i hana au "I worked"); or, perfect participle (i hana "having worked", "who had worked")
e + verb + ana: imperfective aspect (e hana ana au "I was working", "I will be working")
ke + verb + nei: present tense, progressive aspect (ke hana nei au "I am working")
e + verb: future tense/mood (e hana au "I will work"); or, infinitive (e hana "to work"); or, imperative mood (e hana ʻoe "Work thou!")
mai + verb: negative imperative mood (mai hana ʻoe "Do not work thou!")
verb + ʻia: passive voice where the agent is marked by e (Ua hana ʻia ka honua e ka Haku. The world was created by the Creator.)
I also am providing a link to a map of languages with this set of features. Black means "have past tense and have a fully isolating typology," blue means "have past tense, and have both isolating and non-isolating typology, so might fit the bill," and white means "does not fit what you are looking for"
Aspect is another grammatical category which is not quite tense, but similarly is an encoding for a verb in time. Mandarin Chinese uses particles to express aspect, so might also be what you are looking for.